%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: stories.dvi %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o stories.ps stories.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.09.21:1445 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (stories.dvi) @start /Fa 6 117 df69 d100 D<13FE3807FF80380F87C0381E01E0003E13F0EA7C0014F812FCA2B5FCA200FCC7FCA312 7CA2127E003E13186C1330380FC0703803FFC0C6130015167E951A>I<38FF07E0EB1FF8 381F307CEB403CEB803EA21300AE39FFE1FFC0A21A167E951F>110 D<13FE3807FFC0380F83E0381E00F0003E13F848137CA300FC137EA7007C137CA26C13F8 381F01F0380F83E03807FFC03800FE0017167E951C>I<487EA41203A21207A2120F123F B5FCA2EA0F80ABEB8180A5EB8300EA07C3EA03FEEA00F811207F9F16>116 D E /Fb 2 82 df<387F07E0381081801380EA1840EA1420A2EA1210EA1108EA1084A213 421321A213101308A213041302EA280112FEC7FC13157F932B>78 D81 D E /Fc 2 51 df<121812F81218AA12FF080D7D8C0E>49 D<123EEA4180EA80C012 C01200A2EA0180EA03001204EA08401230EA7F8012FF0A0D7E8C0E>I E /Fd 2 85 df<156015E015C0140115801403EC07001406140E140C141C5C1430147014 6014E0495A5C130391C7FC5B130E130C131C13181338133013705B5B12015B120348C8FC 1206120E120C121C5A12301270126012E05A1B2C81AA19>20 D<12C07E12601270123012 387E120C120E120612076C7E12017F12007F1370133013381318131C130C130E7F7F8013 01806D7E146014701430143880140C140E14061407EC0380140115C0140015E015601B2C 81AA19>84 D E /Fe 1 118 df117 D E /Ff 5 111 df<120FEA30C41260EA4068EAC070A21360EA61E0EA3E380E097E8813> 11 D<001FEB0F800007EB1C003805802C144C0009135814983808C11813C200105B13C4 1368137000305B38F861F8190E7E8D1B>77 D<1208A21200A41270129812B01230A21260 126412681270060F7D8E0B>105 D<12381218A35A13C0EA3360EA3440EA7800127E1263 1320EAC340EAC1800B0E7E8D10>107 D110 D E /Fg 17 126 df<132013401380EA01005A1206A25AA25A A212381230A21270A3126012E0AD12601270A31230A212381218A27EA27EA27E7EEA0080 134013200B317A8113>0 D<7E12407E7E12187EA27EA27EA213801201A213C0A3120013 E0AD13C01201A31380A212031300A21206A25AA25A12105A5A5A0B317F8113>I<143014 6014C0EB0180EB03005B130E130C5B1338133013705B5B12015B1203A290C7FC5A120612 0EA2120C121CA312181238A45AA75AB3A31270A77EA41218121CA3120C120EA212061207 7E7FA212017F12007F13701330133813187F130E7F7FEB0180EB00C01460143014637781 1F>18 D<12C012607E7E7E120E7E7E6C7E7F12007F1370133013381318131CA2130C130E 13061307A27F1480A3130114C0A4EB00E0A71470B3A314E0A7EB01C0A414801303A31400 5BA21306130E130C131CA213181338133013705B5B12015B48C7FC5A120E120C5A5A5A5A 14637F811F>I<140C141814381430146014E014C01301EB0380A2EB0700A2130EA25BA2 5BA21378137013F0A25B1201A2485AA4485AA3120F90C7FCA35AA2121EA3123EA4123CA3 127CA81278A212F8B1164B748024>48 D<12C01260127012307E121C120C120E7EA26C7E A26C7EA26C7EA21370A213781338133CA2131C131EA27FA4EB0780A314C01303A314E0A2 1301A314F0A41300A314F8A81478A2147CB1164B7F8024>I<12F8B11278A2127CA8123C A3123EA4121EA3121FA27EA37F1207A36C7EA46C7EA212007FA2137013781338A27FA27F A27FA2EB0380A2EB01C0130014E01460143014381418140C164B748224>64 D<147CB11478A214F8A814F0A31301A414E0A31303A214C0A313071480A3EB0F00A4131E A2131C133CA2133813781370A25BA2485AA2485AA248C7FCA2120E120C121C12185A1270 12605A164B7F8224>I<913801FFE0020F13FC027FEBFF80903A01FF0E3FE0D907F8EB07 F8D90FC0EB00FCD91F00143E013C150F496F7E496F7E48486F7E48486F7E4848167848C7 81000E171C001E171E001C170E003C170F003883A20078188000701703A200F018C04817 01A3BAFCA300E0C7000EC71201A46C170300701880A20078170700381800A2003C5F001C 170E001E171E000E171C000F173C6C6C5E6C6C5E6C6C4B5A6C6C4B5A01784B5A6D4BC7FC 011F153ED90FC014FCD907F8EB07F8D901FFEB3FE06D6CB51280020F01FCC8FC020113E0 3A3A7E7F3F>77 D80 d88 d1850D760127707E71E7C0EEA03E03EF0801FF380E133F130F1301150 E888D13> I< 141814314147014E00C0EB3F0EB3F0EA03FEB45 A53B0130800 0FC7 7 FC150 E818D13> I E/FH 17 17 DF<127812FCA4127806067 D850D>46×D 122×D < 12FCB47 E13E013F813FIFA013F1EB07C0EB01E0EB0F01470148141448 D < 1360EA01E0120 F12FF12F31 203B3A2γ38 7FFF80A2111B7D9A18> II<3838 0180383FFF005B5B5B13C00030C7FCA4EA31F8EA361E38380F80EA3007000013C014E0A3 127812F8A214C012F038600F8038381F00EA1FFEEA07F0131B7E9A18>I<137EEA03FF38 078180380F03C0EA1E07123C387C03800078C7FCA212F813F8EAFB0E38FA0780EAFC0314 C000F813E0A41278A214C0123CEB0780381E0F00EA07FEEA03F8131B7E9A18>I<126038 7FFFE0A214C01480A238E00300EAC0065B5BC65AA25B13E0A212015B1203A41207A66C5A 131C7D9B18>II<77FB512E08881F81007013800601460[0E0147000 0C01430A4000 1400 B03807FFFA21C1C7E9B21>84 D ]101 d104 D<3FF0FC07E9038、E18F3A1F40F20780D980FC13C0A2EB00 F8AB3AFF E7FF3FF8A225127F9128>109 D111μd114×D E/FI 2 82 DF< 39 FFE9F08301408090380400 C0120 CEA0E02EA0D01、380C8080EB4020EB1010EB0808EB0404EB2102EB080EB4080EC40401420A2、14101401404140121123139FFC000 0C0C812401D18080B20>78×D81μE/FJ 25 117 DF11 d14 D<1310A21320A41340EA01F0EA064CEA1846EA3082EA20831260A2EAC1061241EA610C13 18EA3270EA0FC0EA0200A25AA4101A7E9314>30 D<124012E012601220A31240A2128003 097D820A>59 D<1360A213E013C0A212011380A212031300A25A1206A2120E120CA2121C 1218A212381230A212701260A212E05AA20B1D7E9511>61 D<5B5B5B1480130B131B1313 1323A21343138314C0EA0101EA03FFEA02011204120C1208001813E038FE07F815147F93 19>65 D67 d7FC03900EA4858AA4EB78807770A707A700E50E50E5BA4 BA4F88F88F141A147F931 b> I<3907FC1F03900E3603801C06014141480D8038 1 C7FC1386138E139EE67 7138EBA380A080E01C0A26D7EA2478 F38 FF883FC1A147F931 C>75 D 71 D<3907FCF83 d97 D < 123C120 C5AA45 AEA33 80EA 3C60EA3020EA6030A4EAC060A2EA40C0EA6080EA23 012121E0C147F930FF> III1313C13030EA013B0EA0C01210EA30601260A3EAC0C0A21241EA62D0EA3C700 E147E 9311>13131313C136C13C0A3EA07F8EA00 C0EA0180A5EA0300 A51 02021206A21266 12 12E45 A12700 E1A7F9310>102 DI<12061207120612 00A41238124CA2128C12981218A212301232A21264A2123808147F930C>105 D<1330133813301300A4EA01C0EA0260EA0430136012081200A213C0A4EA0180A4EA6300 12E312C612780D1A81930E>I<121E12065AA45A1338135C139CEA3118EA36001238EA3F 80EA61C0EA60C8A3EAC0D013600E147F9312>I<123C120C1218A41230A41260A412C012 C8A312D0126006147F930A>I<3830F87C38590C86384E0D06EA9C0EEA980C1218A24848 5A15801418A23960301900140E190D7F8C1D>I113×D<124120 CA35AAF80EA1800 A25AA45 A1261A212621264 123809127F910D>116 D E/FK 11 11 DF0 d2 D<1204A3EAC460EAF5E0EA3F80EA0E00EA3F80EAF5E0EAC460EA0400A30B0D7E8D11> I<12C012F0123C120E6C7EEA01E0EA0078131C130FEB03C0A2EB0F00131C1378EA01E0EA 0780000EC7FC123C12F012C0C8FCA5B512C0A2121B7D931A>21 D<14101418A280A28080 B612E0A2C7EA030014065CA25CA214101B107E8E21>33 D<1204120EA2121CA31238A212 301270A21260A212C0A2070F7F8F0A>48 D<000F131E393BC06180396060804038403100 D8801A1320130EA3130B3940118040903820C0C03930C07B80390F001E001B0D7E8C21> I62 D<1360B212F14A14147E93A 1A> I<12C0B3AB021D7D950A>106 D E/FL 16 DF<1330A2137813FC139CEA019E13,0EEA020F70413808080801C000 01313E01000303013F002013707137887FFF F8B512FCA216147E931 b> 1 D10 D<120212041208121812101230122012601240A212C0AA1240A21260122012 3012101218120812041202071E7D950D>40 D<1280124012201230121012181208120C12 04A21206AA1204A2120C1208121812101230122012401280071E7E950D>I<1360AAB512 F0A238006000AA14167E9119>43 D<120FEA30C0EA6060A2EA4020EAC030A9EA4020EA60 60A2EA30C0EA0F000C137E9211>48 D<120C121C12EC120CAFEAFFC00A137D9211>I<12 1FEA60C01360EAF07013301260EA0070A2136013C012011380EA02005AEA08101210EA20 20EA7FE012FF0C137E9211>II<136013E0A2EA016012 021206120C120812101220126012C0EAFFFCEA0060A5EA03FC0E137F9211>I<1240EA7F FC13F8EA4010EA80301320EA00401380EA0100A25A12021206A2120EA512040E147E9311 >55 D<387FFFE0B512F0C8FCA6B512F06C13E0140A7E8B19>61 D<39FF0FF0FC39380380 30A2001CEBC020A2001E1460390E04E040A200071480EB0870A23903887900EB9039A238 01F03EEBE01EA23800C00CA31E147F9321>87 D<12F01230A6EA33E0EA3430EA38181230 A9EAFC7E0F147F9312>104 D114×D<1210A312301270EAF80EA3000 0A71380A3EA1100120 E09127F910D>116 D E/FM 25 25 DF0 D<126012F2A12124040C8B0C> I<060C13E083771C038 380801C07EA0E6E5C5AEA03B8EA01F06C5AA247EEA03B8、EA071CEA0EA77E803803807700 1C038 E00E048 136014147A9320> I10 d15 D<9038 7FFF8048 B5FCD807、80C7FC08EC8FC5A12307012126012EA5EA5601212701212877E7E6C7E000 01B5128806C7E90C8FCA700 7FB51280A219227 D92020>18 D(20)D<12C012F0123C0EA03C030C030C030F0133C0EB3F0143C140FEC0380EC0F00 14 3C14F0EB03C010FC7FC133C13F0EA3C000 FC8FC123C12F012C012C9FCA7B61280A219227D9202I24 D<153081A81A1818180ED900C0B712F8A2C912C0ED03×801600 150 65 DA25DA35D25167E942A>33 D49μdI<1460A214E014C01301 1480130314005B1306A2130E130C131C1318133813301370136013E05BA212015B120390 C7FC5A1206120E120CA2121C1218123812301270126012E05AA213287A9D00>54 D<152015E01401A21403A21405A21409A214111431142114611441148113011401130313 06130490380FFFF05BEB3000A25BEA60C0D8718013F8127F48C7127F007E147C003C1400 2020809D21>65 D67 D<150348B512FE000714F8390C00E00048485A1230EA700312F0 00C05B12001307A291C7FCA25BA2130EA2131EA2131CA2133C1338A213781370A213F05B 5B485A20207F9C17>84 D<00C01306B36C130E0060130C0070131C6C1338381F01F03807 FFC03800FE00171A7E981C>91 D<13FE3807FFC0381F01F03838003848131C0060130C00 E0130E481306B3171A7E981C>I<133C13E0EA01C013801203AD13005A121C12F0121C12 077E1380AD120113C0EA00E0133C0E297D9E15>102 D<12F0121C12077E1380AD120113 C0EA00E0133C13E0EA01C013801203AD13005A121C12F00E297D9E15>I<13C0A2120113 8012031300A25A1206120E120CA2121C121812381230A21270126012E05AA27E12601270 1230A212381218121C120CA2120E120612077EA21380120113C01200A20A2A7D9E10>I< 12C0A27E126012701230A212381218121C120CA2120E120612077EA21380120113C01200 A21201138012031300A25A1206120E120CA2121C121812381230A21270126012E05AA20A 2A7E9E10>I<12C0B3B3A502297B9E0C>I<12C0A27E1260A21270123012381218A2121C12 0CA2120E1206A212077EA21380120113C01200A213E01360A213701330A213381318A213 1C130C130E1306A213071303A210297E9E15>110 D<16C015011680150316005D150615 0E150C151C1518153815301570156015E05D14015D140392C7FC000C5B003C1306004E13 0E008E130C0007131C141838038038143014703801C06014E06C6C5A13E1EB7180137301 3BC8FC133F131EA2130CA2222A7E8123>112 D E /Fn 60 123 df<13F8EA030C380E06 04EA1C07383803080030138800701390A200E013A0A214C01480A3EA6007EB0B88383071 90380F80E016127E911B>11 D<38078010EA1FC0383FE020EA7FF038603040EAC0183880 088012001304EB0500A21306A31304A3130CA35BA45BA21320141B7F9115>13 D<1338137FEB87803801030090C7FC7FA27F12007FA2137013F8EA03B8EA063CEA0C1C12 1812381270A212E0A413181338EA6030EA70606C5AEA0F80111E7F9D12>I21 D<3801803000031370A3380700E0A4380E01C0A4381C0388A3EA1E07383E1990383BE0E0 0038C7FCA25AA45AA25A151B7F9119>I<5B1302A45BA45BA2137E3801C980380710E000 0C13600018137000381330EA7020A200E01370A2134014E0A2386041C0EB838038308600 EA1C9CEA07E00001C7FCA41202A414257E9C19>30 D<126012F0A2126004047C830C>58 D<126012F0A212701210A41220A212401280040C7C830C>II<1303A213071306A2130E130C131C1318 A213381330A213701360A213E013C0A21201138012031300A25A1206A2120E120CA2121C 1218A21238123012701260A212E05AA210297E9E15>I<12E01278121EEA0780EA01E0EA 0078131EEB0780EB01E0EB0078141EEC0780A2EC1E001478EB01E0EB0780011EC7FC1378 EA01E0EA0780001EC8FC127812E019187D9520>I<133E13C3380100C0120200031360EA 0780147038030030C7FCA3EBFC70EA0382EA0601120C381800F0003813E01230127014C0 EAE001A21480130314001306A2EA600C6C5AEA1860EA0F80141F7E9D16>64 D<140CA2141CA2143C145CA2149E148EEB010E1302A21304A213081310A2497EEB3FFFEB 40071380A2EA0100A212025AA2001C148039FF803FF01C1D7F9C1F>I<48B5FC39003C01 C090383800E015F01570A25B15F0A2EC01E09038E003C0EC0780EC1F00EBFFFC3801C00F EC0780EC03C0A2EA0380A439070007801500140E5C000E1378B512C01C1C7E9B1F>I<90 3801F80890380E0618903838013890386000F048481370485A48C71230481420120E5A12 3C15005AA35AA45CA300701302A200305B00385B6C5B6C136038070180D800FEC7FC1D1E 7E9C1E>I<48B512F839003C0078013813181510A35BA214081500495AA21430EBFFF038 01C020A4390380404014001580A23907000100A25C1406000E133EB512FC1D1C7E9B1F> 69 D<48B512F038003C00013813301520A35BA214081500495AA21430EBFFF03801C020 A448485A91C7FCA348C8FCA45AEAFFF01C1C7E9B1B>I<903801F80890380E0618903838 013890386000F048481370485A48C71230481420120E5A123C15005AA35AA2EC7FF0EC07 801500A31270140E123012386C131E6C136438070184D800FEC7FC1D1E7E9C21>I<3A01 FFC3FF803A003C00780001381370A4495BA449485AA390B5FC3901C00380A4484848C7FC A43807000EA448131E39FFE1FFC0211C7E9B23>I<3801FFC038003C001338A45BA45BA4 485AA4485AA448C7FCA45AEAFFE0121C7E9B12>I<3A01FFC07F803A003C001E00013813 1815205D5DD97002C7FC5C5C5CEBE04014C0EBE1E013E23801C47013D0EBE03813C0EA03 8080A280EA0700A280A2488039FFE03FF0211C7E9B23>75 D<3801FFE038003C001338A4 5BA45BA4485AA438038002A31404EA0700140C14181438000E13F0B5FC171C7E9B1C>I< D801FE14FFD8003E14F0012EEB01E01502A21504014EEB09C0A201471311A20187EB2380 1543A215833A0107010700A2EB0382A20002EB840E1488A214900004EBA01CA2EB01C012 0C001CEB803C3AFF8103FF80281C7E9B28>I<48B5FC39003C03C090383800E015F01570 A24913F0A315E0EBE001EC03C0EC0700141E3801FFF001C0C7FCA3485AA448C8FCA45AEA FFE01C1C7E9B1B>80 DII3801FFE390013838,E015F01570A24913F0A3EC01E01E013C0EC0780,EC1E00 EBFF03801C038 140EA2EA0380A4380700 1E150 810104130FD8FFE01320 C7EA03C01D1D7E9B20>I<001FB512F0391C03807039300700300020 142012601240130E1280A2000014005BA45BA45BA45BA41201EA7FFF1C1C7F9B18>I<39 7FF03FE0390F000700000E13061404A3485BA4485BA4485BA4485BA35CA249C7FCEA6002 5B6C5AEA1830EA07C01B1D7D9B1C>I<39FFC00FF0391C00038015001402A25C5C121E00 0E5B143014205CA25C49C7FC120FEA07025BA25BA25B5BEA03A013C05BA290C8FCA21C1D 7D9B18>I<3AFFC0FFC0FF3A3C001C003C001C1510143C1620025C1340A2029C1380A290 39011C0100A20102130213045D01085BA2496C5A121ED80E205BA201405B018013C05D26 0F000FC7FCA2000E130EA2000C130CA2281D7D9B27>I<3A01FFC0FF803A001E003C0001 1C13306D13205D010F5B6D48C7FC1482EB038414CCEB01D814F05C130080EB0170EB0278 EB04381308EB103CEB201CEB401EEB800E3801000F00027F1206001E497E39FF803FF021 1C7F9B22>I<39FFE007F8390F0001E0158015006C13026D5A00035BEBC018141000015B 6D5A00005B01F1C7FC13F21376137C1338A25BA45BA4485AEA1FFC1D1C7F9B18>I<90B5 12E09038F001C03901C003809038800700EB000E141E0002131C5C5CC75A495A495A49C7 FC5B131E131C5BEB7002495AEA01C0EA038048485A5A000E1318485B48137048485AB5FC 1B1C7E9B1C>I97 D<123F120 7A2120AA45 AA4EA39 E0EA3A30EA3C181238 127131CA3EA038 A313301313606013C01261EA23 012121E1E1D7E9C12> II1307130FA21306136A61378139CEA010C120 2C13412001 A2 1313A41370A413E0A1E01C01261EAF180EAF300 12E6127C1024809B11> III931F81F044E20C61839 4640 E81CEB80F0EA8F09008E13E09EA2491C01C038 A3157038080771A215E115E23 97 700 70.2664 D8300 31320127 E9124I IA030CαEA0E0618013080138080A138EA88EA3030EA25BEA60185BEA30E0EA0F8011127E 9114>I<3807880380808803D03013E011C01438 1201201A23 8038070A31460380700(E014C0EB0180EB8300 EA8613137890C7FCA25AA4123CB4FC151A819115I)I我< 13F8EI 13C0120 1 A3EA03A4EAF07AA20EA420AA420AA138A21340A218AE0FA00C1A80990FF > I<11C13C0EA07011247 A23 88 70380 A2120EA23 81C0700 A438 180 E20A3EA1E1C1E380C2640γ3807C38013127E9118> II<001CEBC080392701C1C0124714C0398703 8040A2120EA2391C070080A3EC0100EA1806A2381C0E02EB0F04380E13083803E1F01A12 7E911E>I<380787803808C8403810F0C03820F1E0EBE3C03840E1803800E000A2485AA4 3863808012F3EB810012E5EA84C6EA787813127E9118>I<001C13C0EA27011247A23887 0380A2120EA2381C0700A4EA180EA3EA1C1EEA0C3CEA07DCEA001C1318EA6038EAF0305B 485AEA4180003EC7FC121A7E9114>II E /Fo 65 128 df<14FE90380301801306EB0C03EB1C0191C7FC13181338A43803FFFE38 00700EA35CA213E0A25CA3EA01C01472A438038034141891C7FC90C8FCA25A12C612E65A 12781925819C17>12 D<903901FC0FEE9039030E307E0106136090390C0CE07C90391C00 C01C1401A201381438A2EC0380A20003B612F03A00380380701370EC070016E0A313E091 380E01C0A4D801C0EB0388141CA216901501D80380EB00E04A1300A21300143038C63060 38E6384038CC3180D8781FC8FC2725819C25>15 D<121C12261247A2128EA2120E121CA3 1238A21271A31272A2123C08127C910D>I<12031207120E121C1238126012C012800808 729C15>19 D22 D<13031306130813181330136013C0A2EA0180EA 0300A21206A25AA2121C1218A212381230A21270A21260A412E0A51260A5122012301210 7EA2102A7B9E11>40 D<1310A21308130C13041306A51307A51306A4130EA2130CA2131C 1318A213381330A21360A213C0A2EA0180EA0300A212065A5A121012605A102A809E11> I<12181238127812381208A21210A212201240A21280050C7D830D>44 DI<1230127812F0126005047C830D>I<1304130C131813381378 EA07B8EA0070A413E0A4EA01C0A4EA0380A4EA0700A45AEAFFF00E1C7B9B15>49 D<131FEB60C0EBC060EA018038030030A200061360120714C013803803C10013E6EA01FC EA0078EA01BCEA061E487E487E383003801220EA6001A238C00300A21306EA60045BEA38 30EA0FC0141D7D9B15>56 D<133E13E138018180380300C01206120E120C121CA2130112 38A31303001813801307EA080B380C3300EA03C7EA0007130E130C131C1318EAE0305BEA 80C0EAC180003EC7FC121D7C9B15>I<1206120FA212061200AA1230127812F012600812 7C910D>I六十三D<1418A21438A21478A214B8EB0138A2EB023C141C1304130C13081310A21320A2EB 7FFCEBC01C1380EA0100141E0002130EA25A120C001C131EB4EBFFC01A1D7E9C1F>65 D<48B5FC39003C038090383801C0EC00E0A35B1401A2EC03C001E01380EC0F00141EEBFF FC3801C00E801580A2EA0380A43907000F00140E141E5C000E13F0B512C01B1C7E9B1D> I<903803F02090381E0C6090383002E09038E003C03801C001EA038048C7FC000E148012 1E121C123C15005AA35AA41404A35C12705C6C5B00185B6C485AD80706C7FCEA01F81B1E 7A9C1E>I<48B5FC39003C03C090383800E0A21570A24913781538A215785BA4484813F0 A315E03803800115C0140315803907000700140E5C5C000E13E0B512801D1C7E9B1F>I< 48B512F038003C00013813301520A35BA214081500495AA21430EBFFF03801C020A43903 8040801400A2EC0100EA07005C14021406000E133CB512FC1C1C7E9B1C>I<48B512F038 003C00013813301520A35BA214081500495AA21430EBFFF03801C020A448485A91C7FCA3 48C8FCA45AEAFFF01C1C7E9B1B>I<903803F02090381E0C6090383002E09038E003C038 01C001EA038048C7FC000E1480121E121C123C15005AA35AA2903801FF809038001E0014 1CA400705BA27E001813786C139038070710D801F8C7FC1B1E7A9C20>I<3A01FFC3FF80 3A003C007800013131370A495BA40955A5390B5FC3901C0380A448 48 48 C7FCA4380700(0EA448 131E39 FFE1FFC0211C7E9B1F1> I)C5C5C013C013E1E02E02E0EA01 01 C4EBD0701301401401481471EA214080A348 130F39 0FC0211C7E9B20> I < < 3801FFC038 3C000 1338 A45 BA448 5AA438038 03A31404EA0714140C1418148000 0E 13F0B5FC171C7E9B1A> I I<3A01FFC07F803A03C01E00138131815205D5DD97 02C7F.I<3801FFFE39003C038090383801C0EC00E0A3EB7001A315C0EBE0031580EC07 00141C3801FFF001C0C7FCA3485AA448C8FCA45AEAFFE01B1C7E9B1C>I<3801FFFE3900 3C078090383801C015E01400A2EB7001A3EC03C001E01380EC0700141CEBFFE03801C030 80141CA2EA0380A43807003C1520A348144039FFE01E80C7EA0F001B1D7E9B1E>82 DI<001FB512C0381C070138300E0000201480126012405B1280A2000014005B A45BA45BA4485AA41203EA7FFE1A1C799B1E>I<39FF801FC0393C000700001C1304A25C 5CA25CA25C5CA26C48C7FCA213025BA25BA25B5B120F6C5AA25B90C8FCA21206A21A1D77 9B1F>86 D<3AFF83FF07F03A3C007001C00038158002F01300A290380170025D13025D13 045D13085D131001305B1320D81C405BA2D98071C7FCA2381D0072A2001E1374A2001C13 38A20018133014201210241D779B29>I<48B512809038E007003803800EEB001E000213 1C5C485B5C495A1200495A49C7FC130E5BA25B5BEBE00848485AA2EA038048485A120E48 1360003C5BEA380138700780B5FC191C7D9B19>90 D97×D<123F127AA4EA4EA3EA3EA3EA3EA3C038 1270 70130EA3EAE01CA313181338 1330EA60C0EA3180EA1E000 0F1D7C9C13> I < 13F8EA0304EEA1C0EEA181CEA3000 012×70A25AA51 304EA600 8131310EA3060EA0F800 F127C9113> II<13F8EA0704120 CEA1802EA38041230EA7000 8EA7FF0EA000 0A5EA600 4 1308EA30101360EA0F800 F127C9113I112A2138A413A0134E01C0A2EAC180EA180E618767881024819B0D> I < EA0FC01201A244C7FCA4380E07801308EB11C01321381C4180EB8000 0DC7FC12 1EEA3FC0EA13070A2EA7071A31372AE032 EA601C121D7E9C13> I I<13031 30713031300 A71378138CEA010C120 2131C120II91C1E078092663,18C09463A0E084703C008E1380A2120EA23 91C0701C0A3EC0380D8380E1388 A2EC07 0 08151039 701 C0330300 C01C01D127C9122I8EA030CEA0180137080138080A138EA88EA307EA25BEA60185BEA30E0EA0F8011 127C9115> I<38038 7803804C060EB0013E09C01481201201A23 8038070A314603807,00 E014C0EB0180EB8300 EA0613137890C7FCA25AA45 AB4FC151A809115I我< 13F.II035AA31 20EA4EA FFE0EA1CAA35AA4AE4AE080A2EA121266 12380B1A7C990E>I<31C0180EA2E03 03 124EA23 8E0700 A2121CA2EA380EA438 301C80A3EA38 3C38 184D00 EA0F8611127C9116>I<381E0183382703871247148338870701A2120EA2381C0E02A31404 EA180C131C1408EA1C1E380C26303807C3C018127C911C>I<38038780380CC840380870 E012103820E0C014001200A2485AA4EA03811263EAE38212C5EA8584EA787813127E9113 >I<381C0180EA2E03124EA2388E0700A2121CA2EA380EA4EA301CA3EA383CEA1878EA0F B8EA003813301370EAE0605BEA81800043C7FC123C111A7C9114>I127×E/FP 42 122 DF45 d48 D<130E131E137EEA07FE12FFA2 12F81200 B3ABB512FEA31727 7BA622> II<140FA25C5C5C5C5BA2EB03BFEB073F130E131C133C1338137013E0EA01C0EA 038012071300120E5A5A5A12F0B612F8A3C7EA7F00A890381FFFF8A31D277EA622>I<00 181303381F801FEBFFFE5C5C5C14C091C7FC001CC8FCA7EB7FC0381DFFF8381F80FC381E 003F121CC7EA1F8015C0A215E0A21218127C12FEA315C05A0078EB3F80A26CEB7F00381F 01FE6CB45A000313F0C613801B277DA622>III1213123E03FB512F0A31514E015C0A215801939 7000 F5141E5C48 13714785 C648 5A45A5C13079C7FCA25B131E132EA2137EA213 7 7C13FCA41201A8EA00 701C29 7CA822> If00 F9F98F14814F1481414F48141F165A160F145A16071677FA2 90C9F5077123FA26C7E16C6E6C6C141C6C6C143C6C14C876CB4EB01F0903.97FF07C011FB512800 107EBFE9009038 7FF028 29 7CA831>67μd 65 D<9138FE03903907FFFC07011FEFFF0F9039 7F3903907FFF097FF900940FF9009FF001FFF D801FC740848 8048 48 8058A82485 A82127FA290CAFC5AA892B512F8E7F0300 13×0123FA26C7EA26C7E6C7E6C7E6CB6B738 7FF07011FB5129F0107EBF0F9039 00 7FFF32 32 D29 7CA835>71 d I<91387.0078 d82 D<9038FF80600003EBF0E0000F13F8381F80FD383F001F003E1307481303A200FC1301A2 14007EA26C140013C0EA7FFCEBFFE06C13F86C13FE80000714806C14C0C6FC010F13E0EB 007FEC1FF0140F140700E01303A46C14E0A26C13076C14C0B4EB0F80EBE03F39E3FFFE00 00E15B38C01FF01C297CA825>I<007FB71280A39039807F807FD87C00140F00781507A2 0070150300F016C0A2481501A5C791C7FCB3A490B612C0A32A287EA72F>I<3803FF8000 0F13F0381F01FC383F80FE147F801580EA1F00C7FCA4EB3FFF3801FC3FEA0FE0EA1F80EA 3F00127E5AA4145F007EEBDFC0393F839FFC381FFE0F3803FC031E1B7E9A21>97 DB3E1007E133E63E133E63C137C380F C1F8000 1388090C8FC1238 A2123C83FFF88FF6C14C06C14E06C14F0121F38、3C000 077CEB01F848 1300 A47CEB01F0A033FEB 07E0390FC01F806CB8200 38 38 0 7FF0 1 1E28 7E9A22> I I9090FF80F033EBE3F839 0FC1FE1C91F07C7C48 137E00 3EEII.07EA1FC0EA3FE5AE1FC0EA0700 C7 FCA7EAFE0A31 20FB3A3EAFFEA30F2B7EAA12> I3AC1FFC07F0903C08CF81F101FC9090 39 C803F2000 1DF1FE7F01D05BA201E05CFF3FFF8FF00A33 31 B7D9A38>I<38 FFI C07E9038 C1FF809038 C30FC0DC413E0EBC80701D813F013D0A213E0B039 FF3FFFA3201B7D9A25> I 108 D<26FFC07FEB1FC090I<38FFE1FE9038EFFF809038FE0FE0390FF803F09038F001 F801E013FC140015FEA2157FA8157E15FEA215FC140101F013F89038F807F09038FC0FE0 9038EFFF809038E1FC0001E0C7FCA9EAFFFEA320277E9A25>I<90383F80703901FFE0F0 3803F079380FE01D381F800F123FEB00075AA2127E12FEA8127FA27E1380001F130F380F C01F3807F0773801FFE738007F87EB0007A9EC7FFFA320277E9A23>I<38FFC1F0EBC7FC EBC63E380FCC7F13D813D0A2EBF03EEBE000B0B5FCA3181B7F9A1B>I<3803FE30380FFF F0EA3E03EA7800127000F01370A27E00FE1300EAFFE06CB4FC14C06C13E06C13F0000713 F8C6FCEB07FC130000E0137C143C7E14387E6C137038FF01E038E7FFC000C11300161B7E 9A1B>I<13E0A41201A31203A21207120F381FFFE0B5FCA2380FE000AD1470A73807F0E0 000313C03801FF8038007F0014267FA51A>I<39FFE07FF0A3000F1307B2140FA2000713 173903F067FF3801FFC738007F87201B7D9A25>I<39FFFC03FFA3390FF000F0000714E0 7F0003EB01C0A2EBFC0300011480EBFE070000140013FFEB7F0EA2149EEB3F9C14FC6D5A A26D5AA36D5AA26D5AA2201B7F9A23>I<3BFFFC7FFC1FFCA33B0FE00FE001C02607F007 EB0380A201F8EBF00700031600EC0FF801FC5C0001150EEC1FFC2600FE1C5B15FE9039FF 387E3C017F1438EC787F6D486C5A16F0ECE01F011F5CA26D486C5AA2EC800701075CA22E 1B7F9A31>I<39FFFC1FFEA33907F003803803F8079038FC0F003801FE1E00005BEB7F38 14F86D5A6D5A130F806D7E130F497EEB3CFEEB38FFEB787F9038F03F803901E01FC0D803 C013E0EB800F39FFF03FFFA3201B7F9A23>I<39FFFC03FFA3390FF000F0000714E07F00 03EB01C0A2EBFC0300011480EBFE070000140013FFEB7F0EA2149EEB3F9C14FC6D5AA26D 5AA36D5AA26D5AA25CA21307003890C7FCEA7C0FEAFE0E131E131C5BEA74F0EA3FE0EA0F 8020277F9A23>I E /Fq 28 122 df<13FEEA038138060180EA0E03381C010090C7FCA5 B51280EA1C03AE38FF8FF0141A809915>12 D34 d45×D<126012F2A121240407D830B> I < 126012F2A12126012F0A212701210A312> 20A21240A204177D8F0B > 59 D<07FB5FC38 701C07401301A1C014080801300 A300×00 1414B13803FFE0191A7F99 1C> 84 D92 d97 D<12FC121CA913FCEA1D0738 1E038038 1C01 C01300 E0A6EB01C01480131E0300 EA196EA10F8131A8099 15II<133F1307A9EA03E7EA 0C17EA180F147E127012E0A6126012706C5AEA1C37 3807C7E0131A7F99 15> I< 12FC121CA7137DA8131E03、80A2121CAB36FF0141A8099 15> I < 1218123CA612181A612FC121CAE12FF081A809990A> I <12FC121CB3A6EAF800 91A80990A > 108 D < 38 FC7C1F39 1D8E638091 1E0781C0A2×00 1C1301AB39 FF9FE7F81D107F8F20> I我114 dI<1208A41218A21238EAFFC0 EA3800A81320A41218EA1C40EA07800B177F960F>I<38FC1F80EA1C03AB1307120CEA0E 0B3803F3F01410808F15>I<38FF0F80383C0700EA1C061304A26C5AA26C5AA3EA03A0A2 EA01C0A36C5A11107F8F14>I<39FE7F1F8039381C0700003C1306381C0C04130E380E16 081317A238072310149013A33803C1A014E0380180C0A319107F8F1C>I<38FF0F80383C 0700EA1C061304A26C5AA26C5AA3EA03A0A2EA01C0A36C5AA248C7FCA212E112E212E412 7811177F8F14>121 D E /Fr 7 117 df<1303497EA2497EA3EB1BE0A2EB3BF01331A2EB 60F8A2EBE0FCEBC07CA248487EEBFFFE487FEB001F4814800006130FA248EB07C039FF80 3FFCA21E1A7F9921>65 D<12FCA2133F08033E01C03.03C13E0F0A214F8A1214F0A2EB01E003E13C038 3B078038 30FE15151A7E919> I < EA03FCEA01F1EEA1C1F123C1278130EF8C7FCA51278A383C0180131C0300 EA0F06EA03FC11117F9014> 97 d114μdI<1166A42EA2121EEA3FF012FFEA1EAA131318A5 EA0F30EA03E0D187F97 11> I E/FS 40 123 DF12 d34 D<1218123C127124124A21208A21210A21260128060C799 C>0C>39 D<1238 1278A21238 88A21210A21220A21240128050C7D830C>44 DI<1270A212F0126004047C830C>I<120C121E121CA21200 AA1270A212F0126007127C910C>58 D<14201430147014F0A2130180EB0278A21304A213 08801310A21320A2EB403E141EEBFFFEEB801EEA0100141F00027FA25A120C001E148039 FF807FF01C1D7F9C1F>65 D<3807FFFE3900F807809038F001C015E0140015F0A2484813 E0140115C01403EC0780EC1E003803FFFCEBC00FEC0780EC03C0A3EA0780A4EC0780EC0F 00380F001E5CB512E01C1C7F9B1D>I<0007B512E03800F801EBF0001540A4485A142015 00A2146014E048B45A13C01440A3158038078000A2EC0100A25C1406380F000E143EB512 FC1B1C7E9B1C>69 D<903807F01090381C0C3090387002609038C001E03803800048C7FC 000E1460A2481440123C123800781400A35AA3ECFFF0EC0F801407EC0F001270A27EA26C 5B000C131E6C136638038182D8007EC7FC1C1E7C9C21>71 D<3A07FF87FF803A00F800F8 00495BA54848485AA648B55AEBC003A54848485AA6484848C7FC01807F39FFF0FFF0211C 7F9B1F>II<001FB512F0383C07C00030EB80300020142012601240A2EB0F001280000014 00A4131EA65BA65B137C381FFFE01C1C7C9B1E>84 D<3BFFE1FFC0FF803B1F003E003E00 6C011E1318A2161016301620023E1360D9802F13400007014F13C05EEC8F0193C7FCEB81 0F9038C10782EA03C2158413C601C4138813CC01C813D03801F80301F013E0A201E05BA2 01C05B00001301018090C8FC291D7B9B2B>87 D92 dD<123F120、AA3137CEA11381E018012C0133C0131414E0A338 700 1 C0A3EB031400 1306EAF00 CEACC38 EA313E1131D7C9C17 > I < 13FEA0307000 E13800 01C C 1300 EA380690C7FC35AA31260EA700 25BEA300 8EA1C30EA07C011127E9112 I I 97I1414EB7cc3801871C3803008013780120 EA4EB0700、EA0606EA070CEA09F09008C7FC1218A2EF1FFE380FF8014C0EA300 1EA600 014E048 13C0A AA88600 18038 38 0300 EA180EEA07F0161C809215IIi 13C01201A21380C7FCA7EA1F80120 733EA0700 A6120 EA65 A121EEAF800 A1D7F9C0C0> I107μdI<391F8F C0FC39079061063903E0760738078078A2EB0070A4000EEBE00EA64848485A001EEBE01E 3AFF8FF8FF8021127F9124>I<381F8F803807B0C03803C0E0EA0780A21300A4380E01C0 A6381C0380001E13C038FF9FF014127F9117>I<13FCEA0307380E0180001C13C0EA3800 1230007013E0A338E001C0A300601380130338700700EA380EEA1C18EA07E013127E9115 >I<380FC7C03803D8703801E0383803C018EB801C140C140EA33807001CA31438143014 604813C0380EC380EB3E0090C7FCA35AA4B47E171A809117>I114 dI<1202A31206A25A121C123CEAFFE0EA1C00A25AA65A1340A41380A2 EA3100121E0B1A7C9910>I<38FC1F80EA3C07EA1C0338380700A6EA700EA4131EA25BEA 305E381F9F8011127C9117>I<38FF07E0383C03801400EA1C02A25B5B120E5BA25B120F 6C5A13C05B90C7FC7E120213127C9116>I<39FF3FC7E0393C0F038039380E01005C383C 1F02381C1706EB3704EB270CEB6308EB4318381EC390380E83B0000F13A0EB03E06D5A12 0E5C12041B127C911E>I<381FE1FC380781E038038080EBC100EA01C2EA00E613EC1378 13707F137C139CEA011EEA020E487E120C003C138038FE1FF016127F9116>I<380FF07E 3803C038143000011320A214401480EA00E0EBE100A213E213F21374137C137813701330 1320A25BA2EA708000F1C7FC12F312E61278171A809116>I<380FFF80380E0700EA080E 1218EA101C5B5BC65A485AA2EA0382EA0702120E485A1238130CEA7038EAFFF811127F91 12>I E /Ft 92 128 df0 D<1303A2497EA2497E130BEB13E01311EB21F01320497E1478EB807C143C38 01003E141E0002131F8048148014074814C0140348EB01E0003014F000201300006014F8 007FB5FCB612FCA21E1D7E9C23>I<137F3803C1E038070070001C131C003C131E003813 0E0078130F00707F00F01480A50070140000785BA20038130E6C5BA26C5B00061330A200 83EB608000811340A2394180C100007F13FFA3191D7E9C1E>10 DI<137E3801C180EA0301380703C0120EEB018090C7FCA5B512C0EA0E01B0387F87F815 1D809C17>I<90383F07E03901C09C18380380F0D80701133C000E13E00100131892C7FC A5B612FC390E00E01CB03A7FC7FCFF80211D809C23>14 D<90383F07FC3901C0DC1C3903 81F03CEA0701000EEBE01C1300A6B612FC390E00E01CB03A7FC7FCFF80211D809C23>I< 12FC121CB0EAFF8009127F910C>I<120EA2121E1238127012E012800707779C15>19 D<126012F0A71260AD1200A5126012F0A21260041E7C9D0C>33 DI<9038030180A3EB0703 01061300A3EB0E07EB0C06A4EB1C0E007FB512F8B612FC3900301800A3EB7038EB6030A4 B612FC6C14F83901C0E000495AA3EA0381EB0180A3EA0703000690C7FCA31E257E9C23> I<126012F012F812681208A31210A2122012401280050C7C9C0C>39 D<1380EA0100120212065AA25AA25AA35AA412E0AC1260A47EA37EA27EA27E12027EEA00 80092A7C9E10>I<7E12407E12307EA27EA27EA37EA41380AC1300A41206A35AA25AA25A 12205A5A092A7E9E10>I<1306ADB612E0A2D80006C7FCAD1B1C7E9720>43 D<126012F0A212701210A41220A212401280040C7C830C>II<12×6012F0A212600 4047 C830C> I48×D<5A120 7123F12C7120 7B3A5EAFF80D1C7C9B15>I130CA2131C133C13DC139CCEA011C12 031 202120120 C081210121220120124012C0B512C038 01C00 A73801FFC0121C7F9B15E1230130EA312EAE3EAE3EAE8EA88EAF8CY30EAE661307A5 126A2E776EA300 EA3030CEA18C30EA03E0101D7E9B15> I <124087FFF801400 A2EAA24248AA25BA55A1325B1360134013C0A212015BA21203A4120 7A66 ccC7FC111D7E9B15i>I II13F0EA030CEA0404EA0C0EEA181II126012F2A1212601A12126012F2A12124127C910C> I<126012F2A1212601200 AA126012F0A212701210A41220A21240128041A7C910C> I<07FB512C0B612E0C9FCA8πB612E06C14C01B0C7E8F20>>61 D63×D<1306A330FA3EB1780A2EB37 C01323 A2EB43E01341A2EB80F0A33 8010078A2EBFFF838 0 033CA348 7FA2000 C131F800 01E5BB4EBFFF01C1D7F9C1F>65 DI9090118F80618019190807000 07000 E13035A14015AA7781300 A212 7000 0F01400 A800 7014801278A21238 6CEB0100A26C13026C5B380180838 00 E030EB1FC0 191E7E9C1E> IBE0613807000 0137A14015AA77813、00 A2127000、F01400、A0ECFF00EC0F007013071278A21238 7E627、66C130B380180113800、E0609038 1F801C1E7E9C21> I < 39 FFF0F0390F000 0F900AC5B5FCEB00FAD39 FFFFFF F01C1C7F9B1F1> I90901F8080E.7C0133EA13138EA7078EA4070EA30EA0E800F80111D7F9B15> I<3FFF0130F0390F000 0780 EC0600 140C5C5C5C4C7FC13021306130FEB 17801327 EB43C0EB11E013016D7E1478·A280143E141E80158015C09FF03FF01C1C7F9B20> I I<3807FF80800I<3807E080EA1C19EA30051303EA600112E01300A36C 13007E127CEA7FC0EA3FF8EA1FFEEA07FFC61380130FEB07C0130313011280A300C01380 A238E00300EAD002EACC0CEA83F8121E7E9C17>I<007FB512C038700F01006013000040 1440A200C014201280A300001400B1497E3803FFFC1B1C7F9B1E>I<39FFF01FF0390F00 0380EC0100B3A26C1302138000035BEA01C03800E018EB7060EB0F801C1D7F9B1F>I<39 FFE00FF0391F0003C0EC01806C1400A238078002A213C000035BA2EBE00C00011308A26C 6C5AA213F8EB7820A26D5AA36D5AA2131F6DC7FCA21306A31C1D7F9B1F>I<3AFFE1FFC0 FF3A1F003E003C001E013C13186C6D1310A32607801F1320A33A03C0278040A33A01E043 C080A33A00F081E100A39038F900F3017913F2A2017E137E013E137CA2013C133C011C13 38A20118131801081310281D7F9B2B>I<39FFF07FC0390FC01E003807800CEBC0080003 5B6C6C5A13F000005BEB7880137C013DC7FC133E7F7F80A2EB13C0EB23E01321EB40F049 7E14783801007C00027F141E0006131F001F148039FF807FF01C1C7F9B1F>I<39FFF003 FC390F8001E00007EB00C06D13800003EB01006D5A000113026C6C5A13F8EB7808EB7C18 EB3C10EB3E20131F6D5A14C06D5AABEB7FF81E1C809B1F>I<387FFFF0EA7C01007013E0 386003C0A238400780130F1400131E12005B137C13785 BA2485 A120 3EBC010EA0780A2EA 0F048 133000 01E13205A146048 13E080B5FC141C7E9B19 > I < 12FEA212C0B3B312FE A20729 7C9E0C>I < 12FE21206B3B312FEA20729 809E0C>97 D<12FC121CAA137CEA1D838 1E01801C10014060601470A6146014E014C038 1E E 018038 19700 EA10FC141D7F9C17> II<13F8EA018CEA071EA0E0C1300 A6EAFE0EA0E000 B0EA7FEF1F1D809C0D>I<12FC121CAA137C1387EA1D03001E1380121C AD38FF9FF0141D7F9C17>I<1218123CA21218C7FCA712FC121CB0EAFF80091D7F9C0C>I< 13C0EA01E0A2EA00C01300A7EA07E01200B3A21260EAF0C012F1EA6180EA3E000B25839C 0D>I<12FC121CAAEB0FE0EB0780EB06005B13105B5B13E0121DEA1E70EA1C781338133C 131C7F130F148038FF9FE0131D7F9C16>I<12FC121CB3A9EAFF80091D7F9C0C>I<39FC7E 07E0391C838838391D019018001EEBE01C001C13C0AD3AFF8FF8FF8021127F9124>II<3803E080EA0E190185EA3807EA7000 3A2,12E0A61270A2EA38071218EA0E1BEA03E3EA000 03A7EB1FF0141A7F9116II<1204A4120CA2121C123CEAFFE0EA1C00A91310A5120CEA0E20EA03C00C1A7F9910> I<38FC1F80EA1C03AD1307120CEA0E1B3803E3F014127F9117>I<38FF07E0383C038038 1C0100A2EA0E02A2EA0F06EA0704A2EA0388A213C8EA01D0A2EA00E0A3134013127F9116 >I<39FF3FC7E0393C0703C0001CEB01801500130B000E1382A21311000713C4A2132038 03A0E8A2EBC06800011370A2EB8030000013201B127F911E>I<38FF0FE0381E0700EA1C 06EA0E046C5AEA039013B0EA01E012007F12011338EA021C1204EA0C0E487E003C138038 FE1FF014127F9116>I<38FF07E0383C0380381C0100A2EA0E02A2EA0F06EA0704A2EA03 88A213C8EA01D0A2EA00E0A31340A25BA212F000F1C7FC12F312661238131A7F9116>I< EA7FFCEA70381260EA407013F013E0EA41C012031380EA0700EA0F04120E121CEA3C0CEA 380812701338EAFFF80E127F9112>I126 dI E /Fu 11 117 df<127012F8A3127005057C840E>46 D<903807E0109038381830EBE006 3901C0017039038000F048C7FC000E1470121E001C1430123CA2007C14101278A200F814 00A812781510127C123CA2001C1420121E000E14407E6C6C13803901C001003800E002EB 381CEB07E01C247DA223>67 D<3803FFE038001F007FB3A6127012F8A2130EEAF01EEA40 1C6C5AEA1870EA07C013237EA119>74 D80μd97 d101 D<390E1F07F360183801A3E8072010C03C03F03C0E0A2000 000 0E1338 AFF3FF8FFE27 157F942A>109 D < 380E1F8038 F60C038 1E80E0380F00 70 A2120EAF38 FFE7FF18157F941B> I114 D<1202A41206A3120E121E123EEAFFFCEA0E00AB1304A6EA07081203EA01F00E1F7F9E13 >116 D E /Fv 16 118 df<90387F80203801FFE03907C07860380F001C001EEB06E048 130300381301007813001270156012F0A21520A37E1500127C127E7E13C0EA1FF86CB47E 6C13F06C13FCC613FF010F1380010013C0EC1FE01407EC03F01401140015F8A26C1478A5 7E15706C14F015E07E6CEB01C000ECEB038000C7EB070038C1F01E38807FFCEB0FF01D33 7CB125>83 D<13FE380303C0380C00E00010137080003C133C003E131C141EA21208C7FC A3EB0FFEEBFC1EEA03E0EA0F80EA1F00123E123C127C481404A3143EA21278007C135E6C EB8F08390F0307F03903FC03E01E1F7D9E21>97 DII15F0141FA01401307380801C08039008307000 011E201E1300 5A A 2127C1278A212F8A71278A2127C123CA27 E030E13016C13023 8038 00 466C6C48 7E3A00 F0 30 30FF80EB1FC021327 EB125> II<15F090387F03083901C1C41C380380E8 390700700848EB7800001E7FA2003E133EA6001E133CA26C5B6C13706D5A3809C1C0D808 7FC7FC0018C8FCA5121C7E380FFFF86C13FF6C1480390E000FC00018EB01E048EB00F000 701470481438A500701470A26C14E06CEB01C00007EB07003801C01C38003FE01E2F7E9F 21>103 D<120FEA1F80A4EA0F00C7FCABEA0780127FA2120F1207B3A6EA0FC0EAFFF8A2 0D307EAF12>105 D<380780FE39FF83078090388C03C0390F9001E0EA07A06E7E13C0A2 5BB3A2486C487E3AFFFC1FFF80A2211F7E9E25>110 DI<380781FC39FF86078090388801C0390F9000E0D807A0137001C01378497F153E151E15 1FA2811680A716005DA2151E153E153C6D5B01A013705D90389803C0D9860FC7FCEB81F8 0180C8FCAB487EEAFFFCA2212D7E9E25>I<90380FC01090387830303801E00838038004 3907000270481301001E14F0123E003C1300127CA2127812F8A71278127CA2123C123E00 1E13017E6C1302380380043801C0083800F030EB1FC090C7FCAB4A7E91381FFF80A2212D 7E9E23>I<380783E038FF8418EB887CEA0F90EA07A01438EBC000A35BB3487EEAFFFEA2 161F7E9E19>I<3801FC10380E0330381800F048137048133012E01410A37E6C1300127E EA3FF06CB4FC6C13C0000313E038003FF0EB01F813006C133CA2141C7EA27E14186C1338 143000CC136038C301C03880FE00161F7E9E1A>I<1340A513C0A31201A212031207120F 381FFFE0B5FC3803C000B01410A80001132013E000001340EB78C0EB1F00142C7FAB19> II E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 468 482 a Fv(Stories)22 b(ab)r(out)f(groups)g(and)g(sequences) 789 602 y Fu(P)o(eter)16 b(J.)g(Cameron)682 756 y Ft(Sc)o(ho)q(ol)e(of) f(Mathematical)f(Sciences)661 805 y(Queen)k(Mary)d(and)h(W)m (est\014eld)g(College)839 855 y(Mile)g(End)g(Road)833 905 y(London)g(E1)g(4NS)934 955 y(U.K.)428 1046 y Fs(Bey)o(ond)g(Ghor)f (there)i(w)o(as)f(a)g(cit)o(y)m(.)j(All)c(its)h(inhabitan)o(ts)f(w)o (ere)i(blind.)i(A)365 1096 y(king)e(with)g(his)h(en)o(tourage)g(arriv)o (ed)g(near)g(b)o(y)m(.)23 b(He)16 b(brough)o(t)f(his)h(arm)o(y)e(and) 365 1146 y(camp)q(ed)h(in)f(the)h(desert.)22 b(He)16 b(had)e(a)h(migh)o(t)o(y)d(elephan)o(t,)j(whic)o(h)f(he)i(used)f(in)365 1196 y(attac)o(k)f(and)g(to)g(increase)h(the)f(p)q(eople's)g(a)o(w)o (e.)428 1245 y(The)21 b(p)q(opulace)g(b)q(ecame)f(anxious)h(to)f(see)i (the)g(elephan)o(t,)g(and)f(some)365 1295 y(sigh)o(tless)16 b(ones)g(from)e(among)g(this)h(blind)g(comm)o(unit)o(y)d(ran)k(to)f (\014nd)h(it.)22 b(As)365 1345 y(they)10 b(did)f(not)h(ev)o(en)g(kno)o (w)f(the)h(form)e(or)h(shap)q(e)i(of)d(the)j(elephan)o(t)e(they)h(grop) q(ed)365 1395 y(sigh)o(tlessly)m(,)j(gathering)g(information)d(b)o(y)j (touc)o(hing)g(some)f(part)h(of)g(it.)18 b(Eac)o(h)365 1445 y(though)o(t)c(he)g(knew)h(something,)d(b)q(ecause)j(he)g(could)e (feel)h(a)g(part.)428 1494 y(When)9 b(they)h(returned)h(to)e(their)h (fello)o(w-citizens,)f(eager)h(groups)f(clustered)365 1544 y(around)18 b(them.)30 b(Eac)o(h)19 b(of)e(these)j(w)o(as)e (anxious)f(to)h(learn)g(the)h(truth)g(from)365 1594 y(those)d(who)e(w)o (ere)i(themselv)o(es)e(astra)o(y)m(.)20 b(They)15 b(ask)o(ed)g(ab)q (out)f(the)i(form,)c(the)365 1644 y(shap)q(e)j(of)e(the)i(elephan)o(t,) e(and)h(they)h(listened)f(to)g(all)e(they)j(w)o(ere)g(told.)428 1694 y(The)i(man)e(whose)i(hand)f(had)h(reac)o(hed)h(an)e(ear)h(w)o(as) g(ask)o(ed)g(ab)q(out)f(the)365 1743 y(elephan)o(t's)j(nature.)31 b(He)18 b(said:)26 b(\\It)18 b(is)g(a)g(large,)g(rough)g(thing,)g(wide) g(and)365 1793 y(broad,)c(lik)o(e)f(a)g(rug.")428 1843 y(And)i(the)g(one)g(who)g(had)f(felt)h(the)g(trunk)h(said:)j(\\I)c(ha)o (v)o(e)f(the)i(real)e(facts)365 1893 y(ab)q(out)d(it.)16 b(It)11 b(is)f(lik)o(e)f(a)h(straigh)o(t)h(and)f(hollo)o(w)f(pip)q(e,)h (a)o(wful)g(and)g(destructiv)o(e.")428 1943 y(The)16 b(man)e(who)i(had)g(felt)g(its)f(feet)i(and)f(legs)g(said:)22 b(\\It)16 b(is)f(migh)o(t)o(y)f(and)365 1993 y(\014rm,)f(lik)o(e)g(a)g (pillar.")733 2055 y Ft(Mualana)g(Jalaluddin)f(Rumi)g(\(13th)i(cen)o (tury\))h(\(from)d([34)o(]\))893 2142 y Fr(Abstract)423 2207 y Fq(The)k(main)h(theme)g(of)f(this)h(article)g(is)g(that)g(coun)o (ting)h(orbits)f(of)f(an)h(in\014nite)365 2253 y(p)q(erm)o(utation)i (group)f(on)f(\014nite)h(subsets)g(or)f(tuples)h(is)g(v)o(ery)f (closely)i(related)f(to)365 2298 y(com)o(binatorial)i(en)o(umeration;)f (this)e(p)q(oin)o(t)g(of)f(view)h(ties)g(together)f(v)n(arious)i(dis-) 365 2344 y(parate)c(\\stories".)967 2574 y Ft(1)p eop %%Page: 2 2 2 1 bop 262 307 a Fp(1)66 b(Tw)n(o-graphs)22 b(and)h(ev)n(en)g(graphs) 262 398 y Ft(The)c(\014rst)g(story)g(originated)f(with)h(Neil)f (Sloane,)h(when)h(he)f(w)o(as)f(compiling)e(the)k(\014rst)262 448 y(edition)e(of)h(his)g(dictionary)f(of)h(in)o(teger)g(sequences)j ([35)o(].)33 b(He)20 b(observ)o(ed)g(that)f(certain)262 498 y(coun)o(ting)13 b(sequences)j(app)q(eared)f(to)f(agree.)324 548 y(The)j(\014rst)h(sequence)h(en)o(umerates)e Fo(even)h(gr)n(aphs)p Ft(,)f(those)g(in)g(whic)o(h)f(an)o(y)h(v)o(ertex)g(has)262 597 y(ev)o(en)e(v)n(alency)g(\(so)g(that)g(the)h(graph)f(is)g(a)g (disjoin)o(t)f(union)g(of)h(Eulerian)f(graphs\).)22 b(These)262 647 y(graphs)14 b(w)o(ere)g(en)o(umerated)g(b)o(y)g(Robinson)f([29)o(]) g(and)h(Lisk)o(o)o(v)o(ec)g([18)o(].)324 697 y(The)h(second)h(sequence) h(coun)o(ts)f(switc)o(hing)e(classes)j(of)d(graphs.)21 b(If)15 b(\000)g(is)f(a)h(graph)g(on)262 747 y(the)e(v)o(ertex)h(set)g Fn(X)s Ft(,)g(and)f Fn(Y)22 b Ft(is)13 b(a)g(subset)i(of)d Fn(X)s Ft(,)i(the)f(result)h(of)f Fo(switching)f Ft(\000)h(with)g(resp) q(ect)262 797 y(to)e Fn(Y)21 b Ft(is)11 b(obtained)g(b)o(y)h(deleting)f (all)f(edges)j(b)q(et)o(w)o(een)g Fn(Y)21 b Ft(and)11 b(its)g(complemen)o(t,)f(putting)h(in)262 846 y(all)h(edges)k(b)q(et)o (w)o(een)f Fn(Y)23 b Ft(and)14 b(its)g(complemen)o(t)e(whic)o(h)i (didn't)g(exist)g(b)q(efore,H(and)f(LEA)O(VEN)262(896)y((20)B(REST)G(未改变))36 B(Twitc)O(兴)19 B(is)H(an)f(当量)n((a))h(关系)f(on)g(h)(图)g(含)262 946 y(v)o(ErTx)15 b(集)g fn(x)s FT();)f(the)h(equiv)n(alence)f(classes)i (are)e(called)g Fo(switching)h(classes)p Ft(.)j(This)c(concept)262 996 y(w)o(as)f(in)o(tro)q(duced)i(b)o(y)f(Seidel)g([30)o(])f(for)h (studying)f(strongly)h(regular)g(graphs.)324 1046 y(The)20 b(\014nal)g(sequence)i(coun)o(ts)e(t)o(w)o(o-graphs.)36 b(A)20 b Fo(two-gr)n(aph)g Ft(on)f(a)h(set)h Fn(X)j Ft(consists)262 1095 y(of)16 b(a)g(set)i Fm(T)27 b Ft(of)16 b Fo(triples)f Ft(or)i(3-elemen)o(t)f(subsets)i(of)e Fn(X)21 b Ft(with)16 b(the)h(prop)q(ert)o(y)h(that)f(an)o(y)f(4-)262 1145 y(elemen)o(t)g(subset)j(of)e Fm(T)28 b Ft(con)o(tains)17 b(an)g(ev)o(en)h(n)o(um)o(b)q(er)f(of)g(elemen)o(ts)g(of)g Fm(T)10 b Ft(.)29 b(Tw)o(o-graphs)262 1195 y(w)o(ere)19 b(in)o(tro)q(duced)g(b)o(y)f(G.)f(Higman)g(in)h(a)g(construction)h(of)f (Con)o(w)o(a)o(y's)f(third)h(sp)q(oradic)262 1245 y(group.)23 b(The)16 b(theory)g(has)g(b)q(een)h(dev)o(elop)q(ed)f(in)f(man)o(y)f (directions:)23 b(Seidel)15 b(has)h(written)262 1295 y(sev)o(eral)f(surv)o(eys)h([31)o(],)e([33)o(],)h([32)o(].)20 b(They)15 b(also)g(link)f(sev)o(eral)h(themes)g(in)g(com)o(binatorics,) 262 1345 y(including)g(equiangular)g(lines)h(in)f(Euclidean)i(space,)g (and)f(double)g(co)o(v)o(ers)h(of)e(complete)262 1394 y(graphs.)324 1444 y(It)e(w)o(as)h(already)f(kno)o(wn)g(that)g(switc)o (hing)h(classes)g(and)g(t)o(w)o(o-graphs)f(are)h(equin)o(umer-)262 1494 y(ous.)19 b(There)c(is)f(a)g(map)f(from)f(graphs)j(on)f(the)h(set) g Fn(X)j Ft(to)c(t)o(w)o(o-graphs)g(on)g Fn(X)s Ft(,)g(as)h(follo)o (ws:)262 1544 y(the)d(triples)h(of)e(the)i(t)o(w)o(o-graph)e(are)i(all) e(3-sets)i(whic)o(h)f(con)o(tain)g(an)f(o)q(dd)i(n)o(um)o(b)q(er)e(of)h (edges)262 1594 y(of)j(the)h(graph.)24 b(Ev)o(ery)16 b(t)o(w)o(o-graph)f(is)h(obtained)g(in)f(this)h(w)o(a)o(y)m(,)e(and)i (graphs)g(\000)1532 1600 y Fl(1)1566 1594 y Ft(and)g(\000)1675 1600 y Fl(2)262 1643 y Ft(giv)o(e)e(the)h(same)f(t)o(w)o(o-graph)g(if)g (and)g(only)g(if)g(they)h(lie)f(in)g(the)i(same)e(switc)o(hing)g (class.)21 b(So)262 1693 y(there)15 b(is)e(a)h(natural)f(bijection)h (from)e(switc)o(hing)i(classes)h(to)f(t)o(w)o(o-graphs.)324 1743 y(It)e(w)o(as)h(also)e(kno)o(wn)h(that)h(switc)o(hing)f(classes)i (and)e(ev)o(en)h(graphs)g(on)f(an)g(o)q(dd)g(n)o(um)o(b)q(er)262 1793 y(of)j(v)o(ertices)j(are)f(equin)o(umerous.)25 b(\(An)o(y)16 b(switc)o(hing)g(class)h(on)f(an)g(o)q(dd)h(n)o(um)o(b)q(er)f(of)f(v)o (er-)262 1843 y(tices)h(con)o(tains)g(a)f(unique)h(ev)o(en)h(graph,)e (obtained)h(b)o(y)g(taking)e(an)o(y)i(graph)f(in)h(the)g(class)262 1892 y(and)f(switc)o(hing)h(with)g(resp)q(ect)j(to)d(the)g(set)h(of)f (v)o(ertices)h(of)f(o)q(dd)g(degree.\))26 b(But)17 b(no)f(suc)o(h)262 1942 y(corresp)q(ondence)f(exists)d(if)f(the)i(n)o(um)o(b)q(er)e(of)h (v)o(ertices)h(is)f(ev)o(en.)18 b(Mallo)o(ws)11 b(and)h(Sloane)f([21)o (])262 1992 y(pro)o(v)o(ed)16 b(that)h(the)h(n)o(um)o(b)q(ers)e(w)o (ere)i(equal)e(b)o(y)h(deriving)f(a)h(form)o(ula)d(for)j(the)g(n)o(um)o (b)q(er)f(of)262 2042 y(switc)o(hing)h(classes)i(and)f(observing)f (that)h(it)g(coincides)g(with)g(the)g(Robinson{Lisk)o(o)o(v)o(ec)262 2092 y(form)o(ula)11 b(for)i(the)i(n)o(um)o(b)q(er)e(of)g(ev)o(en)i (graphs.)324 2142 y(The)f(\\righ)o(t")f(explanation)g([6)o(])g (actually)g(sho)o(ws)i(that)e(the)i(classes)g(are)f(dual.)k(Let)c Fn(X)262 2191 y Ft(b)q(e)k(a)f(set)h(of)f Fn(n)h Ft(p)q(oin)o(ts,)g (and)f Fn(V)27 b Ft(the)18 b(set)g(of)f(all)g(graphs)g(on)h(the)g(v)o (ertex)g(set)h Fn(X)s Ft(.)29 b(Eac)o(h)262 2241 y(graph)15 b(can)i(b)q(e)g(represen)o(ted)h(b)o(y)e(a)g(binary)g(v)o(ector)h(of)e (length)h Fn(n)p Ft(\()p Fn(n)11 b Fm(\000)g Ft(1\))p Fn(=)p Ft(2)k(whose)i(ones)262 2291 y(giv)o(e)12 b(the)i(p)q(ositions)f (of)g(the)h(edges.)19 b(So)13 b Fn(V)23 b Ft(is)13 b(a)g(v)o(ector)i (space)f(o)o(v)o(er)f(GF\(2\))g(of)g(dimension)262 2341 y Fn(n)p Ft(\()p Fn(n)c Fm(\000)h Ft(1\))p Fn(=)p Ft(2.)20 b(The)15 b(addition)e(in)i Fn(V)24 b Ft(corresp)q(onds)16 b(to)f(taking)f(the)h(symmetric)e(di\013erence)262 2391 y(of)g(the)h(edge)h(sets)g(of)e(the)i(t)o(w)o(o)e(graphs.)18 b(W)m(e)c(consider)h(t)o(w)o(o)e(subsets)j(of)d Fn(V)d Ft(:)967 2574 y(2)p eop %%Page: 3 3 3 2 bop 324 307 a Fm(\017)20 b Fn(U)5 b Ft(,)14 b(the)g(set)h(of)e (complete)g(bipartite)h(graphs;)324 390 y Fm(\017)20 b Fn(W)6 b Ft(,)14 b(the)g(set)h(of)e(ev)o(en)i(graphs.)262 473 y(It)f(is)g(easy)g(to)g(see)h(that)g Fn(U)j Ft(is)c(a)g(subspace)i (of)d Fn(V)d Ft(,)j(spanned)i(b)o(y)f(the)h(stars.)k(No)o(w)14 b(a)g(graph)262 523 y(is)g(ev)o(en)h(if)f(and)g(only)g(if)g(it)g(is)g (orthogonal)g(to)g(all)f(stars;)j(so)e Fn(W)19 b Ft(=)13 b Fn(U)1352 508 y Fk(?)1380 523 y Ft(,)h(and)g Fn(W)21 b Ft(is)14 b(also)g(a)262 573 y(subspace.)324 623 y(The)h(cosets)h(of)e Fn(U)19 b Ft(in)14 b Fn(V)24 b Ft(are)15 b(precisely)h(the)f(switc)o (hing)f(classes)i(of)e(graphs.)20 b(So)14 b Fn(V)5 b(=U)262 672 y Ft(is)13 b(the)i(set)g(of)e(switc)o(hing)h(classes.)19 b(Since)c Fn(W)i Ft(=)12 b Fn(U)1061 657 y Fk(?)1089 672 y Ft(,)h(this)h(quotien)o(t)g Fn(V)5 b(=U)18 b Ft(is)c(isomorphic) 262 722 y(to)j(the)h(dual)f(space)i Fn(W)647 707 y Fk(\003)684 722 y Ft(of)e Fn(W)6 b Ft(,)18 b(not)f(just)h(as)g(v)o(ector)h(space,)g (but)f(as)f(mo)q(dule)g(for)g(the)262 772 y(symmetric)10 b(group)i(on)g Fn(X)s Ft(.)18 b(No)o(w)12 b(a)g(group)g(acting)g(on)g (a)g(\014nite)h(v)o(ector)g(space)g(has)g(equally)262 822 y(man)o(y)18 b(orbits)j(on)f(the)h(space)g(and)f(on)h(its)f(dual,)h (b)o(y)f(Brauer's)i(lemma)17 b([4)o(];)23 b(and)d(the)262 872 y(orbits)14 b(of)h(the)g(symmetric)e(group)i(are)g(the)h (isomorphism)11 b(classes.)23 b(So)14 b(the)i(n)o(um)o(b)q(ers)e(of)262 922 y(switc)o(hing)f(classes)i(and)f(ev)o(en)h(graphs)f(are)g(equal.) 324 971 y(Recen)o(tly)m(,)j(I)f(noticed)h(another)h(feature,)f(whic)o (h)g(ma)o(y)e(b)q(e)i(related)h(in)e(some)g(w)o(a)o(y)g(to)262 1021 y(this)g(dualit)o(y)m(.)26 b(As)17 b(noted)g(ab)q(o)o(v)o(e,)g(an) g(ev)o(en)g(graph)g(is)g(the)g(disjoin)o(t)f(union)g(of)h(Eulerian)262 1071 y(graphs.)34 b(A)19 b(similar-lo)q(o)o(king)d(decomp)q(osition)i (holds)h(for)f(t)o(w)o(o-graphs.)34 b(W)m(e)19 b(de\014ne)h(a)262 1121 y(relation)13 b Fm(\030)h Ft(on)g(the)h(p)q(oin)o(t)e(set)i(of)e (a)h(t)o(w)o(o-graph)f(b)o(y)h(the)h(rule)f(that)g Fn(x)d Fm(\030)h Fn(y)k Ft(if)d(and)h(only)f(if)262 1171 y(either)i Fn(x)c Ft(=)i Fn(y)j Ft(or)e(no)g(triple)g(con)o(tains)g Fn(x)g Ft(and)g Fn(y)q Ft(.)20 b(F)m(rom)13 b(the)h(de\014nition)g(of)g (a)g(t)o(w)o(o-graph,)262 1220 y(it)h(is)g(easy)h(to)f(see)h(that)g (this)f(is)g(an)h(equiv)n(alence)f(relation,)g(and)g(is)g(ev)o(en)h(a)f (congruence,)262 1270 y(that)c(is,)h(mem)o(b)q(ership)e(of)h(a)h (triple)f(in)h Fm(T)21 b Ft(is)12 b(una\013ected)h(if)e(w)o(e)h (replace)h(some)e(of)g(its)h(p)q(oin)o(ts)262 1320 y(b)o(y)g(equiv)n (alen)o(t)g(ones.)18 b(Th)o(us,)12 b(a)g(t)o(w)o(o-graph)g(is)h (describ)q(ed)h(b)o(y)e(a)g(partition)g(of)g Fn(X)s Ft(,)h(with)f(no) 262 1370 y(structure)i(on)d(the)i(parts)g(of)e(the)i(partition,)e(and)h (the)g(structure)i(of)e(a)f Fo(r)n(e)n(duc)n(e)n(d)h Ft(t)o(w)o(o-graph)262 1420 y(\(one)18 b(in)f(whic)o(h)g(all)g Fm(\030)p Ft(-classes)i(are)f(singletons\))f(on)h(the)g(set)h(of)e (parts.)29 b(\(By)18 b(con)o(trast,)262 1469 y(for)c(ev)o(en)i(graphs,) f(w)o(e)h(ha)o(v)o(e)f(an)g(Eulerian)g(graph)f(on)h(eac)o(h)h(part)f (of)g(the)h(partition,)e(and)262 1519 y(no)k(structure)i(on)f(the)g (set)g(of)f(parts;)j(this)e(is,)g(in)f(some)f(v)n(ague)h(sense,)j (\\dual")d(to)g(the)262 1569 y(preceding.\))324 1619 y(The)11 b(n)o(um)o(b)q(ers)f(of)f(Eulerian)i(graphs)f(and)g(of)g (reduced)i(t)o(w)o(o-graphs)e(on)g Fn(n)h Ft(p)q(oin)o(ts)f(agree)262 1669 y(for)j Fn(n)f Fm(\024)f Ft(4)j(but)g(di\013er)g(for)g Fn(n)d Ft(=)h(5.)262 1806 y Fp(2)66 b(Groups)23 b(and)g(coun)n(ting)262 1897 y Ft(Let)c Fn(G)f Ft(b)q(e)h(a)g(p)q(erm)o(utation)e(group)h(on)h (a)f(set)i(\012.)32 b(Usually)18 b(\012)h(will)e(b)q(e)i(in\014nite.)32 b(The)262 1947 y(group)16 b Fn(G)g Ft(is)g(said)g(to)g(b)q(e)h Fo(oligomorphic)f Ft(if)f(the)i(n)o(um)o(b)q(er)f(of)f(orbits)i(of)e Fn(G)h Ft(on)g(the)h(set)g(of)262 1997 y Fn(n)p Ft(-subsets)e(of)e (\012)g(is)h(\014nite)g(for)f(ev)o(ery)h(p)q(ositiv)o(e)g(in)o(teger)g Fn(n)p Ft(.)k(\(More)c(ab)q(out)f(the)h(deriv)n(ation)262 2046 y(of)f(this)h(term)g(b)q(elo)o(w.\))19 b(So)14 b(ev)o(ery)h (\014nite)f(p)q(erm)o(utation)f(group)h(is)g(oligomorphic.)i(If)e Fn(G)g Ft(is)262 2096 y(oligom)o(orphic,)c(w)o(e)j(let)g Fn(f)658 2102 y Fj(n)681 2096 y Ft(\()p Fn(G)p Ft(\))g(\(or)g(just)g Fn(f)926 2102 y Fj(n)949 2096 y Ft(,)g(if)f(the)h(group)g(is)g(clear\)) g(denote)h(the)f(n)o(um)o(b)q(er)262 2146 y(of)g(orbits)h(of)f Fn(G)h Ft(on)f Fn(n)p Ft(-sets.)324 2196 y(Design)k(theorists)h(will)e (recognise)j(this)e(set-up.)29 b(Supp)q(ose)18 b(that)g(w)o(e)f(w)o(an) o(t)g(to)h(con-)262 2246 y(struct)d(a)f Fn(t)p Ft(-design)g(on)g(\012)g (with)g(blo)q(c)o(k)f(size)i Fn(k)g Ft(admitting)d(the)i(group)g Fn(G)p Ft(.)19 b(Let)14 b Fn(T)1538 2252 y Fl(1)1557 2246 y Fn(;)7 b(:)g(:)g(:)e(;)i(T)1674 2252 y Fj(a)262 2295 y Ft(b)q(e)j(the)h(orbits)g(on)f Fn(t)p Ft(-sets,)h(and)f Fn(K)781 2301 y Fl(1)800 2295 y Fn(;)d(:)g(:)g(:)e(;)i(K)928 2301 y Fj(b)954 2295 y Ft(the)k(orbits)g(on)f Fn(k)q Ft(-sets,)h(where)h Fn(a)f Ft(=)h Fn(f)1531 2301 y Fj(t)1546 2295 y Ft(,)e Fn(b)i Ft(=)f Fn(f)1661 2301 y Fj(k)1682 2295 y Ft(.)262 2345 y(No)o(w)i(w)o(e)i(build)e(a)h(collapsed)g (incidence)h(matrix)d Fn(M)17 b Ft(=)12 b(\()p Fn(m)1205 2351 y Fj(ij)1235 2345 y Ft(\))i(of)f(size)i Fn(a)10 b Fm(\002)f Fn(b)p Ft(,)14 b(where)h Fn(m)1664 2351 y Fj(ij)262 2395 y Ft(is)e(the)h(n)o(um)o(b)q(er)e(of)h Fn(k)q Ft(-sets)i(in)d(the)i Fn(j)r Ft(th)g(orbit)f(whic)o(h)h(con)o (tain)f(a)g(\014xed)g Fn(t)p Ft(-set)i(from)c(the)j Fn(i)p Ft(th)262 2445 y(orbit.)25 b(No)o(w)16 b(the)h(game)e(is)h(to)g(select) i(a)e(subset)i(of)d(the)i(columns)f(of)f Fn(M)21 b Ft(suc)o(h)d(that)e (the)967 2574 y(3)p eop %%Page: 4 4 4 3 bop 262 307 a Ft(submatrix)11 b(has)h(constan)o(t)i(ro)o(w)e(sums;) g(then)h(the)g(union)f(of)g(the)i(corresp)q(onding)f(orbits)g(is)262 357 y(the)h(blo)q(c)o(k)g(set)g(of)g(the)g(design.)324 407 y(This)j(do)q(esn't)h(w)o(ork)f(if)g(\012)h(is)f(in\014nite,)h (since)g(the)g(n)o(um)o(b)q(ers)f Fn(m)1354 413 y Fj(ij)1402 407 y Ft(ma)o(y)e(b)q(e)j(in\014nite.)262 457 y(Ho)o(w)o(ev)o(er,)d (collapsing)g(the)h(matrix)d(the)j(other)g(w)o(a)o(y)f(do)q(es)h(mak)o (e)e(sense:)23 b(let)16 b Fn(P)j Ft(=)c(\()p Fn(p)1637 463 y Fj(ij)1666 457 y Ft(\),)262 506 y(where)f Fn(p)402 512 y Fj(ij)445 506 y Ft(is)f(the)h(n)o(um)o(b)q(er)e(of)h Fn(t)p Ft(-sets)i(in)e(the)h Fn(i)p Ft(th)g(orbit)f(whic)o(h)g(are)h (con)o(tained)f(in)g(a)g(\014xed)262 556 y Fn(k)q Ft(-set)g(from)e(the) i Fn(j)r Ft(th)h(orbit.)j(W)m(e)c(will)e(return)j(to)f(this)g(later;)f (but,)h(unfortunately)m(,)f(I)h(ha)o(v)o(e)262 606 y(nothing)g(more)g (to)g(sa)o(y)h(ab)q(out)g(constructing)h(designs!)324 656 y(The)i(concept)h(whic)o(h)f(links)f(this)h(kind)g(of)f(orbit)g (coun)o(ting)h(to)g(com)o(binatorial)c(en)o(u-)262 706 y(meration)h(is)h(that)h(of)f(a)g(homogeneous)g(relational)f (structure.)26 b(A)16 b Fo(r)n(elational)f(structur)n(e)262 756 y Fn(X)g Ft(on)d(\012)g(consists)i(of)d(a)h(n)o(um)o(b)q(er)g(of)f (relations)h(on)g Fn(X)k Ft(of)c(v)n(arious)f(arities.)18 b(Th)o(us,)12 b(man)o(y)e(of)262 805 y(our)i(fa)o(v)o(ourite)g (structures)j(\(graphs,)e(digraphs,)f(tournamen)o(ts,)g(total)g(or)g (partial)g(orders,)262 855 y(t)o(w)o(o-graphs\))18 b(are)h(relational.) 31 b(An)19 b Fo(induc)n(e)n(d)h(substructur)n(e)e Ft(of)h(a)f (relational)f(structure)262 905 y(on)f(a)h(subset)h(of)f(\012)g(is)g (obtained)f(b)o(y)h(simply)e(taking)h(the)i(restrictions)g(of)e(all)g (the)i(rela-)262 955 y(tions)11 b(to)h(this)g(subset.)18 b(No)o(w)12 b Fn(X)j Ft(is)d Fo(homo)n(gene)n(ous)h Ft(if)e(ev)o(ery)i (isomorphism)8 b(b)q(et)o(w)o(een)13 b(\014nite)262 1005 y(substructures)j(of)e Fn(X)j Ft(can)d(b)q(e)h(extended)g(to)f(an)g (automorphism)c(of)k Fn(X)s Ft(.)324 1054 y(The)j(classical)f(example)f (of)h(a)h(homogeneous)e(structure)k(is)d(the)i(rational)d(n)o(um)o(b)q (ers)262 1104 y Fi(Q)h Ft(as)j(ordered)i(set.)34 b(Giv)o(en)19 b(an)o(y)g(t)o(w)o(o)f Fn(n)p Ft(-sets)j(of)d(rationals,)h(arranged)h (in)e(increasing)262 1154 y(order)j(as)f Fn(a)456 1160 y Fl(1)497 1154 y Fn(<)i(a)573 1160 y Fl(2)614 1154 y Fn(<)g Fm(\001)7 b(\001)g(\001)21 b Fn(<)i(a)816 1160 y Fj(n)858 1154 y Ft(and)e Fn(b)964 1160 y Fl(1)1004 1154 y Fn(<)i(b)1077 1160 y Fl(2)1117 1154 y Fn(<)g Fm(\001)7 b(\001)g(\001)21 b Fn(<)h(b)1315 1160 y Fj(n)1337 1154 y Ft(,)g(there)f(is)g(a)f(unique)262 1204 y(isomorphism)12 b(b)q(et)o(w)o(een)18 b(the)e(substructures)q(,)j(taking)c Fn(a)1166 1210 y Fj(i)1195 1204 y Ft(to)h Fn(b)1266 1210 y Fj(i)1296 1204 y Ft(for)g Fn(i)f Ft(=)g(1)p Fn(;)7 b(:)g(:)g(:)e(;)i(n)p Ft(.)24 b(This)262 1254 y(can)14 b(b)q(e)h(extended)h(to)e(an)h(order-preserving)h(map)d(on)h(all)f(the) i(rationals)f(b)o(y)g(\\\014lling)e(in")262 1303 y(the)i(in)o(terv)n (als)f(\()p Fn(a)537 1309 y Fj(i)551 1303 y Fn(;)7 b(a)592 1309 y Fj(i)p Fl(+1)647 1303 y Ft(\))14 b(with)f(linear)h(maps,)e(and)h (translating)g(the)h(t)o(w)o(o)g(ends)g(suitably)m(.)324 1353 y(Based)f(on)f(this)h(example,)e(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)13 b([13)o(])f(ga)o(v)o(e)g(a)g(necessary)i(and)e (su\016cien)o(t)h(condition)262 1403 y(for)g(a)g(class)h Fm(C)i Ft(of)d(\014nite)h(structures)i(to)d(b)q(e)h(all)f(the)h (\014nite)g(substructures)i(of)d(a)h(coun)o(table)262 1453 y(homogeneous)c(structure.)20 b(I)11 b(will)f(giv)o(e)h(only)g(a)h (brief)f(description)h(of)f(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e's) 12 b(condition)262 1503 y(here)17 b(\(it)e(is)h(discussed)i(in)e (detail)f(in)g([7]\).)24 b(It)16 b(is)g(required)h(that)f Fm(C)i Ft(is)e(closed)g(under)h(iso-)262 1553 y(morphism;)11 b(closed)k(under)g(taking)e(induced)i(substructures;)i(con)o(tains)e (only)e(coun)o(tably)262 1602 y(man)o(y)d(structures)k(up)f(to)e (isomorphism;)e(and)j(has)h(the)f Fo(amalgamation)i(pr)n(op)n(erty)e Ft(\(whic)o(h)262 1652 y(asserts)17 b(that,)f(giv)o(en)g(t)o(w)o(o)f (structures)k Fn(B)921 1658 y Fl(1)940 1652 y Fn(;)7 b(B)990 1658 y Fl(2)1024 1652 y Fm(2)14 b(C)19 b Ft(with)c(a)h(common)d (substructure)19 b Fn(A)p Ft(,)262 1702 y(there)c(is)f(a)h(structure)h Fn(C)f Fm(2)d(C)17 b Ft(in)d(whic)o(h)g Fn(B)947 1708 y Fl(1)980 1702 y Ft(and)g Fn(B)1092 1708 y Fl(2)1126 1702 y Ft(can)g(b)q(oth)h(b)q(e)g(em)o(b)q(edded,)f(so)h(that)262 1752 y(their)f(in)o(tersection)i(is)e(at)h(least)f Fn(A)p Ft(\).)20 b(The)15 b(\014rst)g(three)h(conditions)e(are)h(usually)f(ob) o(vious,)262 1802 y(but)g(the)g(amalgam)o(atio)o(n)d(prop)q(ert)o(y)k (ma)o(y)d(require)j(more)e(e\013ort)i(to)f(v)o(erify)m(.)j(Man)o(y)d (famil-)262 1851 y(iar)c(classes)i(of)e(\014nite)h(structures)i (\(graphs,)e(tournamen)o(ts,)g(p)q(osets,)h(triangle-free)f(graphs,)262 1901 y(t)o(w)o(o-graphs,)f(.)d(.)f(.)h(\))17 b(satisfy)11 b(the)h(condition,)e(and)h(man)o(y)e(others)j(\(bipartite)f(graphs,)g (trees,)262 1951 y(.)6 b(.)g(.)h(\))27 b(can)17 b(b)q(e)g(made)f(to)h (satisfy)f(it)g(after)h(small)e(mo)q(di\014cation.)24 b(F)m(or)17 b(example,)e(graphs)262 2001 y(with)e(a)h(\014xed)g (bipartition)f(satisfy)g(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e's)14 b(conditions.)324 2051 y(No)o(w)d(let)g Fn(X)k Ft(b)q(e)d(a)f (homogeneous)f(structure,)k(and)d Fm(C)i Ft(the)f(class)g(of)e(its)i (\014nite)f(substruc-)262 2100 y(tures.)18 b(If)11 b Fn(G)h Ft(is)f(the)i(automorphism)8 b(group)k(of)f Fn(X)s Ft(,)h(then)h Fn(G)p Ft(-orbits)e(on)g Fn(n)p Ft(-sets)i(corresp)q(ond) 262 2150 y(to)e(isomorphism)e(classes)k(of)e Fn(n)p Ft(-elemen)o(t)h (structures)i(in)d Fm(C)k Ft(\(unlab)q(elled)c(substructures)k(of)262 2200 y Fn(X)s Ft(\).)j(Moreo)o(v)o(er,)12 b(giv)o(en)g(an)o(y)g(p)q (erm)o(utation)f(group)h(on)f(a)h(coun)o(table)h(set,)g(it)e(is)h(p)q (ossible)h(to)262 2250 y(construct)f(a)e(structure)i(on)e(whic)o(h)h (the)g(group)f(acts)h(\\homogeneously".)k(So)10 b(the)h(problem)262 2300 y(of)h(calculating)g(the)h(n)o(um)o(b)q(ers)g Fn(f)774 2306 y Fj(n)796 2300 y Ft(\()p Fn(G)p Ft(\))g(for)g(oligomo)o(rphic)d (groups)k Fn(G)e Ft(is)h(iden)o(tical)f(to)h(that)262 2350 y(of)g(en)o(umerating)g(unlab)q(elled)h(structures)j(in)c(a)h (class)h(satisfying)e(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e's)14 b(condition)g(\(a)262 2399 y Fo(F)m(r)n(a)l(\177)-17 b(\020ss)o(\023)d(e)14 b(class)p Ft(,)f(I)h(will)e(sa)o(y)i(for)f (short\).)324 2449 y(The)19 b(term)g(\\oligom)o(orphic")d(is)j(deriv)o (ed)h(from)d(\\few)i(shap)q(es",)i(and)e(is)g(c)o(hosen)h(to)967 2574 y(4)p eop %%Page: 5 5 5 4 bop 262 307 a Ft(express)15 b(this)f(relationship)f(b)q(et)o(w)o (een)j(the)e(group)g(orbits)g(and)f(the)i(isomorphism)10 b(classes)262 357 y(of)16 b(structures)j(\(\\shap)q(es"\))f(in)e(a)h (class)g(with)g(only)f(\014nitely)g(man)o(y)f(of)i(an)o(y)f(giv)o(en)g (\014nite)262 407 y(size)e(\(\\few"\).)262 544 y Fp(3)66 b(An)23 b(inequalit)n(y)j(and)d(a)f(Ramsey)g(problem)262 635 y Ft(Because)c(of)e(the)h(connection)h(describ)q(ed)g(in)e(the)h (last)g(section,)g(an)o(y)g(general)f(result)i(on)262 685 y(orbit)g(n)o(um)o(b)q(ers)h(for)g(oligomorphi)o(c)e(groups)i(is)g (a)g(metatheorem)f(ab)q(out)h(en)o(umerating)262 735 y(structures)j(in)d(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)20 b(classes.)36 b(The)20 b(most)f(basic)h(result)g(of)f(this)h(kind)f(is) g(that)h(the)262 784 y(n)o(um)o(b)q(ers)13 b Fn(f)449 790 y Fj(n)486 784 y Ft(are)h(non-decreasing:)19 b Fn(f)873 790 y Fj(n)907 784 y Fm(\024)12 b Fn(f)971 790 y Fj(n)p Fl(+1)1036 784 y Ft(.)324 834 y(This)20 b(w)o(as)g(pro)o(v)o(ed)g(for)g (\014nite)g(p)q(erm)o(utation)f(groups)i(b)o(y)f(Livingstone)f(and)h(W) m(ag-)262 884 y(ner)f([19)o(],)g(using)f(c)o(haracter)i(theory)f(of)f (the)h(symmetric)e(group.)31 b(This)19 b(result)g(can)g(b)q(e)262 934 y(translated)14 b(in)o(to)f(a)h(pro)q(of)f(using)h(Blo)q(c)o(k's)g (lemma)d(together)k(with)e(the)i(fact)f(that)g(the)g(re-)262 984 y(duced)e(incidence)h(matrices)e(de\014ned)i(in)f(the)g(last)g (section)g(ha)o(v)o(e)g(full)e(rank)i(pro)o(vided)g(that)262 1034 y Fm(j)p Ft(\012)p Fm(j)i(\025)h Fn(t)c Ft(+)g Fn(k)q Ft(.)24 b(As)17 b(men)o(tioned)e(there,)i(the)g(matrix)d Fn(P)22 b Ft(is)16 b(meaningful)d(ev)o(en)k(when)g(\012)f(is)262 1083 y(in\014nite,)e(and)h(can)g(b)q(e)g(sho)o(wn)g(to)g(ha)o(v)o(e)g (full)e(rank,)i(from)e(whic)o(h)i(the)g(inequalit)o(y)f(can)h(b)q(e)262 1133 y(deduced)g(\(taking)e Fn(t)f Ft(=)f Fn(n)p Ft(,)j Fn(k)e Ft(=)g Fn(n)d Ft(+)h(1\).)324 1183 y(A)19 b(second,)h (completely)e(di\013eren)o(t)i(pro)q(of)f(w)o(as)f(found)h(b)o(y)f(P)o (ouzet)i([25)o(],)f(based)h(on)262 1233 y(Ramsey's)8 b(Theorem.)17 b(The)10 b(essen)o(tial)h(ingredien)o(t)g(can)f(b)q(e)h (stated)h(as)e(a)g(Ramsey)f(theorem)262 1283 y(as)k(follo)o(ws:)262 1374 y Fh(Theorem)h(3.1)21 b Fo(Supp)n(ose)16 b(that)f Fn(t)c Fm(\024)h Fn(k)q Fo(,)j(and)g(let)g(the)f Fn(t)p Fo(-subsets)h(of)g(the)g(in\014nite)g(set)g Ft(\012)g Fo(b)n(e)262 1424 y(p)n(artitione)n(d)e(into)h(\014nitely)g(many)h (classes)f Fn(T)963 1430 y Fj(i)991 1424 y Fo(\()p Ft(1)d Fm(\024)h Fn(i)g Fm(\024)g Fn(a)p Fo(\),)h(al)r(l)h(non-empty.)20 b(F)m(or)13 b(any)i Fn(k)q Fo(-)262 1474 y(set)10 b Fn(U)5 b Fo(,)11 b(let)g Fn(p)455 1480 y Fj(i)468 1474 y Ft(\()p Fn(U)5 b Ft(\))11 b Fo(denote)h(the)e(numb)n(er)h(of)g Fn(t)p Fo(-subsets)g(of)g Fn(U)k Fo(in)c(the)g(class)f Fn(T)1411 1480 y Fj(i)1425 1474 y Fo(.)18 b(L)n(et)10 b Fn(P)17 b Ft(=)12 b(\()p Fn(p)1648 1480 y Fj(ij)1678 1474 y Ft(\))262 1523 y Fo(b)n(e)h(the)h(matrix)g(whose)g(c)n(olumns)g (ar)n(e)g(the)g(distinct)f(ve)n(ctors)h Ft(\()p Fn(p)1252 1529 y Fl(1)1270 1523 y Ft(\()p Fn(U)5 b Ft(\))p Fn(;)i(:)g(:)g(:)e(;)i (p)1449 1529 y Fj(a)1469 1523 y Ft(\()p Fn(U)e Ft(\)\))1550 1508 y Fk(>)1592 1523 y Fo(which)262 1573 y(o)n(c)n(cur.)32 b(Then,)21 b(after)e(r)n(e-or)n(dering)f(r)n(ows)h(and)h(c)n(olumns)g (if)f(ne)n(c)n(essary,)i(the)e(matrix)g Fn(P)262 1623 y Fo(is)f(upp)n(er)i(triangular)e(with)g(non-zer)n(o)i(diagonal)f (\(that)h(is,)f Fn(p)1255 1629 y Fj(ij)1303 1623 y Ft(=)h(0)f Fo(for)f Fn(i)i(>)f(j)r Fo(,)h(while)262 1673 y Fn(p)283 1679 y Fj(ii)319 1673 y Fm(6)p Ft(=)12 b(0)p Fo(\).)324 1764 y Ft(Lik)o(e)f(all)g(go)q(o)q(d)h(Ramsey)e(theorems,)i(this)g(one) g(has)g(a)g(\014nite)g(v)o(ersion)g(as)g(w)o(ell:)k(it)c(holds)262 1814 y(if)17 b(\012)h(is)g(su\016cien)o(tly)h(large)f(in)g(terms)g(of)f Fn(t;)7 b(k)q(;)g(a)p Ft(.)30 b(Here)20 b(the)f(pro)q(of)f(giv)o(es)g (\\su\016cien)o(tly)262 1864 y(large")c(as)h(a)g(v)n(ast,)g(iterated)g (Ramsey)f(n)o(um)o(b)q(er;)h(y)o(et)g(there)h(is)f(some)f(evidence)j (that)e(the)262 1914 y(result)g(holds)f(for)g(sets)i(of)e(quite)g(mo)q (dest)g(size.)21 b(Nob)q(o)q(dy)14 b(kno)o(ws)g(the)i(true)f(v)n(alue)f (of)g(this)262 1963 y(Ramsey)e(function.)324 2013 y(Note)k(that)g(the)h (fact)f(that)g(the)h(ro)o(ws)f(of)f Fn(P)22 b Ft(are)16 b(linearly)f(indep)q(enden)o(t)i(is)f(a)g(simple)262 2063 y(consequence)i(of)e(the)h(Ramsey)e(theorem,)h(and)h(the)g (inequalit)o(y)e(follo)o(ws)f(directly)m(.)26 b(\(W)m(e)262 2113 y(tak)o(e)16 b(the)h(classes)h(of)e Fn(t)p Ft(-sets)h(to)g(b)q(e)g (the)g(orbits)g(of)e Fn(G)p Ft(.)26 b(No)o(w)16 b(t)o(w)o(o)g Fn(k)q Ft(-sets)i(giving)d(rise)i(to)262 2163 y(di\013eren)o(t)d (columns)f(lie)g(in)g(di\013eren)o(t)i(orbits,)e(so)h Fn(f)1054 2169 y Fj(k)1089 2163 y Ft(is)f(at)h(least)g(equal)f(to)h (the)g(n)o(um)o(b)q(er)f(of)262 2212 y(distinct)h(columns,)e(whic)o(h)i (is)f(at)h(least)g(the)h(n)o(um)o(b)q(er)e Fn(f)1136 2218 y Fj(t)1165 2212 y Ft(of)g(ro)o(ws.\))324 2262 y(Macpherson,)22 b(in)e([20)o(])g(and)g(other)g(pap)q(ers,)j(has)d(pro)o(v)o(ed)g(some)g (p)q(o)o(w)o(erful)f(results)262 2312 y(ab)q(out)11 b(the)h(rate)g(of)f (gro)o(wth)h(of)f(the)h(sequence)i(\()p Fn(f)1033 2318 y Fj(n)1056 2312 y Ft(\()p Fn(G)p Ft(\)\).)j(F)m(or)11 b(example,)f(if)h Fn(G)g Ft(is)h(primitiv)o(e)262 2362 y(\(that)17 b(is,)f(preserv)o(es)k(no)c(non-trivial)f(equiv)n(alence)i (relation\),)g(then)g(either)h Fn(f)1520 2368 y Fj(n)1543 2362 y Ft(\()p Fn(G)p Ft(\))e(=)h(1)262 2412 y(for)c(all)g Fn(n)p Ft(,)g(or)h(the)g(sequence)i(gro)o(ws)e(at)g(least)g(exp)q(onen) o(tially)m(.)967 2574 y(5)p eop %%Page: 6 6 6 5 bop 262 307 a Fp(4)66 b(Direct)23 b(and)g(wreath)f(pro)r(ducts)262 398 y Ft(Next)e(w)o(e)g(turn)h(to)e(t)o(w)o(o)h(metho)q(ds)f(of)h (constructing)h(new)f(groups)g(from)e(old.)36 b(If)20 b(our)262 448 y(groups)f(are)g(automorphism)d(groups)j(of)g (homogeneous)f(structures,)k(then)e(these)g(t)o(w)o(o)262 498 y(constructions)e(translate)g(in)o(to)f(op)q(erations)g(on)h(the)g (\014nite)f(substructures,)k(and)c(hence)262 548 y(on)11 b(the)h(sequences)j(en)o(umerating)10 b(them.)17 b(These)c(op)q (erations)f(are)g(quite)f(general,)h(and)g(do)262 597 y(not)e(dep)q(end)i(on)f(ha)o(ving)f(a)g(group)h(around.)17 b(\(This)11 b(p)q(oin)o(t)f(is)h(the)g(heart)h(of)e(the)i(philosoph)o (y)262 647 y(of)h(these)k(notes.)j(In)15 b(fact,)f(a)h(com)o (binatorial)c(setting)k(more)f(general)h(than)f(group)h(orbits)262 697 y(has)j(b)q(een)h(dev)o(elop)q(ed)f(b)o(y)g(A.)f(Jo)o(y)o(al)g([16) o(])g(and)h(his)g(sc)o(ho)q(ol,)g(under)h(the)f(name)f Fo(sp)n(e)n(cies)p Ft(.)262 747 y(This)c(is)h(v)o(ery)g(close)h(in)e (spirit)h(to)g(what)f(I)h(am)e(doing)h(here.\))324 797 y(The)18 b(op)q(erations)g(on)f(sequences)j(can)e(often)g(b)q(e)g (expressed)i(concisely)e(in)f(terms)g(of)262 846 y(their)d(generating)g (functions.)k(Accordingly)m(,)13 b(if)g Fn(G)g Ft(is)h(oligomorphic,)c (w)o(e)15 b(let)789 957 y Fn(f)809 963 y Fj(G)837 957 y Ft(\()p Fn(t)p Ft(\))d(=)955 905 y Fk(1)941 918 y Fg(X)940 1005 y Fj(n)p Fl(=0)1009 957 y Fn(f)1029 963 y Fj(n)1052 957 y Ft(\()p Fn(G)p Ft(\))p Fn(t)1132 940 y Fj(n)1155 957 y Fn(:)262 1075 y Ft(\(Note)i(that)g Fn(f)488 1081 y Fl(0)507 1075 y Ft(\()p Fn(G)p Ft(\))d(=)h(1,)h(since)i(there)g(is)f (a)f(unique)h(empt)o(y)f(set.\))324 1124 y(First,)g(let's)h(ha)o(v)o(e) f(a)g(couple)h(of)e(groups)i(to)f(feed)h(in)o(to)f(the)h (constructions.)19 b(Let)14 b Fn(S)i Ft(de-)262 1174 y(note)11 b(the)g(symmetric)e(group)h(on)h(an)f(in\014nite)h(set,)h (and)e Fn(A)h Ft(the)g(group)g(of)f(order-preserving)262 1224 y(p)q(erm)o(utations)h(of)g(the)i(rational)e(n)o(um)o(b)q(ers.)17 b(Then)c Fn(f)1092 1230 y Fj(n)1115 1224 y Ft(\()p Fn(S)r Ft(\))f(=)g Fn(f)1250 1230 y Fj(n)1273 1224 y Ft(\()p Fn(A)p Ft(\))g(=)g(1)g(for)f(all)g Fn(n)p Ft(.)18 b(\(This)262 1274 y(is)11 b(clear)i(for)f Fn(S)r Ft(,)g(and)g(follo)o(ws)f(for)g Fn(A)h Ft(from)f(our)h(pro)q(of)f(of)h(the)h(homogeneit)o(y)d(of)h Fi(Q)p Ft(.\))k(Hence)262 1324 y Fn(f)282 1330 y Fj(S)306 1324 y Ft(\()p Fn(t)p Ft(\))k(=)g Fn(f)443 1330 y Fj(A)471 1324 y Ft(\()p Fn(t)p Ft(\))g(=)g(1)p Fn(=)p Ft(\(1)12 b Fm(\000)g Fn(t)p Ft(\).)31 b(The)19 b(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)19 b(class)f(corresp)q(onding)i(to)e Fn(S)j Ft(consists)e(of)262 1374 y(\014nite)14 b(sets)h(without)f(an)o (y)g(additional)e(structure;)k(that)e(for)g Fn(A)g Ft(consists)h(of)f (\014nite)g(totally)262 1423 y(ordered)h(sets.)k(In)14 b(eac)o(h)g(case,)h(there)g(is)f(just)g(one)g(ob)r(ject)g(of)g(eac)o(h) g(size)h Fn(n)p Ft(.)324 1473 y(Let)h Fn(H)i Ft(b)q(e)e(a)f(p)q(erm)o (utation)f(group)h(on)g(a)g(set)i(\000,)e(and)g Fn(K)j Ft(a)e(p)q(erm)o(utation)e(group)h(on)262 1523 y(\001.)i(The)12 b Fo(dir)n(e)n(ct)h(pr)n(o)n(duct)f Fn(H)d Fm(\002)d Fn(K)16 b Ft(\(the)d(set)g(of)f(all)f(ordered)j(pairs)e(\()p Fn(h;)7 b(k)q Ft(\))12 b(with)g Fn(h)f Fm(2)h Fn(H)j Ft(and)262 1573 y Fn(k)e Fm(2)g Fn(K)s Ft(,)i(with)f(p)q(oin)o(t)o (wise)g(op)q(erations\))i(acts)f(on)f(the)i(disjoin)o(t)e(union)g(of)g (the)h(sets)h(\000)f(and)262 1623 y(\001,)i(where)h(the)g(\014rst)g (comp)q(onen)o(t)f(of)f(a)h(pair)g(acts)h(on)f(\000)g(and)g(the)h (second)g(comp)q(onen)o(t)262 1672 y(acts)13 b(on)f(\001.)17 b(No)o(w)12 b(a)h(\014nite)f(subset)i(of)e(\000)7 b Fm([)g Ft(\001)k(has)i(the)g(form)e(\000)1249 1678 y Fl(0)1274 1672 y Fm([)c Ft(\001)1344 1678 y Fl(0)1361 1672 y Ft(,)12 b(where)i(\000)1530 1678 y Fl(0)1561 1672 y Ft(and)e(\001)1675 1678 y Fl(0)262 1722 y Ft(are)k(\014nite)h(subsets)h(of)e(\000)g(and)g (\001)g(resp)q(ectiv)o(ely;)j(t)o(w)o(o)d(suc)o(h)h(sets)h(lie)d(in)h (the)h(same)f(orbit)262 1772 y(of)e Fn(H)e Fm(\002)e Fn(K)18 b Ft(if)c(and)g(only)g(if)g(their)h(in)o(tersections)h(with)e (\000)h(lie)f(in)g(the)h(same)f Fn(H)s Ft(-orbit,)g(and)262 1822 y(similarly)d(for)k(\001)f(and)h Fn(K)s Ft(.)20 b(So)15 b(the)g(sequence)i(\()p Fn(f)1038 1828 y Fj(n)1061 1822 y Ft(\()p Fn(H)c Fm(\002)d Fn(K)s Ft(\)\))15 b(is)g(the)g Fo(c)n(onvolution)h Ft(of)e(the)262 1872 y(sequences)i(\()p Fn(f)486 1878 y Fj(n)509 1872 y Ft(\()p Fn(H)s Ft(\)\))e(and)g(\()p Fn(f)726 1878 y Fj(n)749 1872 y Ft(\()p Fn(K)s Ft(\)\):)682 1985 y Fn(f)702 1991 y Fj(n)725 1985 y Ft(\()p Fn(H)e Fm(\002)e Fn(K)s Ft(\))i(=)959 1933 y Fj(n)939 1945 y Fg(X)942 2034 y Fj(i)p Fl(=0)1006 1985 y Fn(f)1026 1991 y Fj(i)1040 1985 y Ft(\()p Fn(H)s Ft(\))p Fn(f)1130 1991 y Fj(n)p Fk(\000)p Fj(i)1191 1985 y Ft(\()p Fn(K)s Ft(\))p Fn(;)262 2100 y Ft(and)k(the)h(generating)f(functions)h(simply)d(m)o (ultiply:)20 b Fn(f)1148 2106 y Fj(H)r Fk(\002)p Fj(K)1251 2100 y Ft(=)c Fn(f)1319 2106 y Fj(H)1351 2100 y Fn(f)1371 2106 y Fj(K)1403 2100 y Ft(.)26 b(Note)17 b(that)f(the)262 2150 y(terms)11 b(of)g(the)h(sequence)h(\()p Fn(f)694 2156 y Fj(n)717 2150 y Ft(\()p Fn(H)8 b Fm(\002)t Fn(S)r Ft(\)\))13 b(are)f(the)g(partial)e(sums)h(of)g(the)h(sequence)i(\()p Fn(f)1573 2156 y Fj(n)1596 2150 y Ft(\()p Fn(H)s Ft(\)\).)324 2200 y(More)h(imp)q(ortan)o(tly)m(,)c(w)o(e)k(see)g(that)g(a)f (structure)i(in)e(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)g(for)f Fn(H)e Fm(\002)e Fn(K)18 b Ft(is)262 2250 y(just)13 b(the)g(disjoin)o(t)f(union)h(of)f(structures)j(for)e Fn(H)j Ft(and)d Fn(K)s Ft(.)18 b(So)12 b(the)i(direct)g(pro)q(duct)g (of)e(p)q(er-)262 2300 y(m)o(utation)g(groups)i(corresp)q(onds)j(to)d (the)h(disjoin)o(t)e(union)h(of)g(com)o(binatorial)e(structures.)262 2350 y(F)m(or)g(example,)g(the)i(ob)r(jects)g(in)f(the)g(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)14 b(class)g(for)e Fn(S)f Fm(\002)d Fn(S)16 b Ft(can)e(b)q(e)f(tak)o(en)h(to)f(b)q(e)g(\014nite) 262 2399 y(sets)h(whose)f(elemen)o(ts)g(are)h(coloured)f(red)h(and)f (blue;)g(and)g Fn(f)1218 2405 y Fj(n)1241 2399 y Ft(\()p Fn(S)d Fm(\002)e Fn(S)r Ft(\))k(=)g Fn(n)c Ft(+)g(1,)k(since)i(an)262 2449 y Fn(n)p Ft(-set)g(can)g(con)o(tain)g(0)p Fn(;)7 b Ft(1)p Fn(;)g Ft(2)p Fn(;)g(:)g(:)f(:)t(;)h(n)13 b Ft(blue)h(elemen)o(ts.)967 2574 y(6)p eop %%Page: 7 7 7 6 bop 865 465 a Ft(\000)977 302 y(\001)898 328 y Fg(z)p 917 328 59 5 v 59 w(}|)p 1014 328 V 59 w({)p 905 573 2 237 v 948 573 V 991 454 a Fm(\001)7 b(\001)g(\001)p 1081 573 V 659 668 a Ft(Figure)14 b(1:)k(\000)9 b Fm(\002)h Ft(\001)j(as)h(a)f(co)o(v)o(ering)h(of)f(\001)324 800 y(There)e(is)f(another)g(w)o(ell-kno)o(wn)f(p)q(erm)o(utation)g(action) h(of)f(the)i(direct)g(pro)q(duct,)g(on)f(the)262 850 y(Cartesian)15 b(pro)q(duct)g(of)g(the)g(sets)h(\000)f(and)g(\001:)20 b(the)15 b(pair)g(\()p Fn(h;)7 b(k)q Ft(\))14 b(maps)g(\()p Fn(\015)r(;)7 b(\016)r Ft(\))15 b(to)g(\()p Fn(\015)r(h;)7 b(\016)r(k)q Ft(\).)262 900 y(\(This)13 b(is)g(the)h Fo(pr)n(o)n(duct)g(action)g Ft(of)e Fn(H)f Fm(\002)d Fn(K)s Ft(.\))19 b(If)12 b Fn(H)k Ft(and)e Fn(K)i Ft(are)e(oligomo)o (rphic,)d(then)i(so)h(is)262 950 y Fn(H)e Fm(\002)e Fn(K)17 b Ft(in)d(this)g(action.)19 b(Ho)o(w)o(ev)o(er,)c(the)g(n)o(um)o(b)q (er)e(of)h(orbits)g(on)g Fn(n)p Ft(-sets)i(is)e(not)g(uniquely)262 1000 y(determined)e(b)o(y)h(the)h(corresp)q(onding)g(n)o(um)o(b)q(ers)e (for)h Fn(H)j Ft(and)d Fn(K)s Ft(.)k(\()p Fo(Exer)n(cise:)h Ft(c)o(hec)o(k)c(that,)262 1050 y(in)h(the)i(pro)q(duct)g(action,)f Fn(f)703 1056 y Fl(2)722 1050 y Ft(\()p Fn(S)e Fm(\002)d Fn(S)r Ft(\))16 b(=)f(3,)h(while)g Fn(f)1106 1056 y Fl(2)1125 1050 y Ft(\()p Fn(A)11 b Fm(\002)g Fn(A)p Ft(\))k(=)h(4.\))25 b(There)17 b(are)f(some)262 1099 y(v)o(ery)e(in)o(teresting)g (questions)h(here,)f(but)g(I)g(w)o(on't)f(sa)o(y)h(an)o(y)f(more)g(ab)q (out)h(this.)324 1149 y(The)j(other)h(construction)g(is)f(the)g Fo(wr)n(e)n(ath)g(pr)n(o)n(duct)g Ft(of)f(p)q(erm)o(utation)g(groups.) 28 b(It)17 b(is)262 1199 y(con)o(v)o(enien)o(t)g(to)h(build)e(up)i(the) g(action)f(\014rst.)29 b(The)18 b(group)f Fn(G)g Ft(=)h Fn(H)10 b Ft(W)m(r)c Fn(K)21 b Ft(acts)d(on)f(the)262 1249 y(set)g(\000)10 b Fm(\002)h Ft(\001;)17 b(but)f(the)h(factors)g (should)f(not)g(b)q(e)h(regarded)g(as)f(ha)o(ving)f(the)i(same)e (status.)262 1299 y(Rather,)c(think)f(of)h(\000)s Fm(\002)s Ft(\001)g(as)g(the)g(disjoin)o(t)f(union)g(of)h Fm(j)p Ft(\001)p Fm(j)e Ft(copies)j(of)e(\000,)h(eac)o(h)g(cop)o(y)g(indexed) 262 1348 y(b)o(y)16 b(a)g(p)q(oin)o(t)h(of)f(\001,)g(as)h(in)f(Figure)h (1.)26 b(\(F)m(ormally)m(,)14 b(the)j(cop)o(y)g(\000)1285 1354 y Fj(\016)1320 1348 y Ft(of)f(\000)g(indexed)i(b)o(y)e Fn(\016)j Ft(is)262 1398 y Fm(f)p Ft(\()p Fn(\015)r(;)7 b(\016)r Ft(\))18 b(:)f Fn(\015)j Fm(2)e Ft(\000)p Fm(g)p Ft(.\))29 b(In)17 b(top)q(ological)f(terms,)i(w)o(e)g(regard)g(\000)11 b Fm(\002)h Ft(\001)18 b(as)f(a)h(co)o(v)o(ering)f(of)g(\001)262 1448 y(whose)d Fo(\014br)n(es)g Ft(are)g(the)g(sets)i(\000)740 1454 y Fj(\016)758 1448 y Ft(,)d(eac)o(h)i(isomorphic)d(to)i(\000.)324 1498 y(The)i Fo(b)n(ase)h(gr)n(oup)f Fn(B)j Ft(of)c(the)i(wreath)f(pro) q(duct)h(consists)g(of)e(all)g(p)q(erm)o(utations)g(built)262 1548 y(from)10 b Fm(j)p Ft(\001)p Fm(j)h Ft(indep)q(enden)o(tly)h(c)o (hosen)h(elemen)o(ts)f(of)g Fn(H)s Ft(,)g(eac)o(h)g(acting)g(on)g(the)h (corresp)q(onding)262 1597 y(\014bre.)18 b(It)13 b(is)f(a)h(cartesian)g (pro)q(duct)h(of)e Fm(j)p Ft(\001)p Fm(j)f Ft(copies)j(of)e Fn(H)s Ft(.)17 b(The)c Fo(top)h(gr)n(oup)f Fn(T)19 b Ft(is)12 b(the)i(group)262 1647 y Fn(K)s Ft(,)f(p)q(erm)o(uting)f(the)i (\014bres)h(b)o(y)e(acting)g(on)h(their)g(indices)g(according)f(to)g (its)h(giv)o(en)f(action)262 1697 y(on)h(\001.)19 b(The)c(wreath)g(pro) q(duct)g(is)g(no)o(w)f(the)h(pro)q(duct)g Fn(B)r(T)6 b Ft(.)20 b(\(In)15 b(group-theoretic)g(terms,)262 1747 y Fn(B)i Ft(is)d(normalised)f(b)o(y)h Fn(T)20 b Ft(and)15 b Fn(B)d Fm(\\)d Fn(T)19 b Ft(=)13 b(1,)g(so)i(the)g(wreath)g(pro)q (duct)h(is)e(the)h(semi-direct)262 1797 y(pro)q(duct)f(of)g Fn(B)i Ft(b)o(y)e Fn(T)6 b Ft(.\))324 1847 y(What)12 b(do)g(the)h(orbits)g(of)f Fn(H)d Ft(W)m(r)e Fn(K)15 b Ft(on)e Fn(n)p Ft(-sets)g(lo)q(ok)f(lik)o(e?)17 b(Eac)o(h)12 b Fn(n)p Ft(-set)i(is)e(partitioned)262 1896 y(b)o(y)18 b(its)i(in)o(tersections)g(with)f(the)h(\014bres;)i(these)f(in)o (tersections)g(can)e(b)q(e)h(indep)q(enden)o(tly)262 1946 y(p)q(erm)o(uted)14 b(to)h(an)o(y)f(other)i(sets)g(in)f(the)g (same)f(\014bre)i(b)o(y)e(the)i(base)f(group.)21 b(Ho)o(w)o(ev)o(er,)15 b(the)262 1996 y(w)o(a)o(y)f(in)h(whic)o(h)g(the)h(set)g(of)f(parts)g (of)g(the)h(partition)e(is)h(p)q(erm)o(uted)h(b)o(y)f(the)h(top)f (group)g(is)262 2046 y(less)f(easy)g(to)g(describ)q(e.)324 2096 y(Supp)q(ose)20 b(that)e Fn(H)k Ft(and)d Fn(K)j Ft(are)d(automorphism)c(groups)20 b(of)e(homogeneous)g(struc-)262 2145 y(tures.)j(Then)16 b(an)e Fn(n)p Ft(-elemen)o(t)g(structure)j(in)d (the)i(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)g(for)f Fn(H)c Ft(W)m(r)d Fn(K)17 b Ft(consists)f(of)262 2195 y(a)11 b(partition)h(of)g(the)g(p)q(oin)o(t)g(set,)h(together)g(with)f (indep)q(enden)o(tly)h(c)o(hosen)h(structures)g(from)262 2245 y(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)h(for)f Fn(H)i Ft(on)e(eac)o(h)h(part)f(of)f(the)i(partition,)e (and)h(a)g(structure)i(from)c(the)262 2295 y(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)14 b(class)g(for)g Fn(K)j Ft(on)c(the)i(set)g(of)e(parts.)324 2345 y(This)h(com)o(binatorial)e (\\comp)q(osition",)g(as)j(with)g(the)g(disjoin)o(t)f(union)g(for)g (the)i(direct)262 2394 y(pro)q(duct,)11 b(is)f(meaningful)e(ev)o(en)j (if)e(there)j(are)e(no)g(groups)h(around.)16 b(Consider)11 b(the)g(example)262 2444 y(in)j(the)i(\014rst)f(section.)22 b(The)16 b(class)f(of)g(ev)o(en)g(graphs)g(is)g(the)h(comp)q(osition)d (of)h(the)i(class)f(of)967 2574 y(7)p eop %%Page: 8 8 8 7 bop 262 307 a Ft(Eulerian)12 b(graphs)g(with)g(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)12 b(class)h(for)f Fn(S)r Ft(;)h(while)f(the)h(class)f(of)g(t)o(w)o(o-graphs)g(is)g(the)262 357 y(comp)q(osition)g(of)h(the)i(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)f(for)g Fn(S)j Ft(with)d(the)g(class)h (of)f(reduced)h(t)o(w)o(o-graphs.)k(\(If)262 407 y(there)14 b(w)o(ere)f(homogeneous)f(structures)k(for)c(the)i(relev)n(an)o(t)e (classes,)i(with)f(automorphism)262 457 y(groups)h Fn(E)r(v)q(en)p Ft(,)g Fn(E)r(ul)q(er)q(ian)p Ft(,)g Fn(T)6 b(w)q(oGr)14 b Ft(and)g Fn(RedT)6 b(w)q(oGr)q Ft(,)12 b(then)j(w)o(e)f(w)o(ould)f (ha)o(v)o(e)466 543 y Fn(E)r(v)q(en)g Fm(\030)f Fn(E)r(ul)q(er)q(ian)7 b Ft(W)m(r)f Fn(S;)90 b(T)6 b(w)q(oGr)12 b Fm(\030)g Fn(S)e Ft(W)m(r)c Fn(RedT)g(w)q(oGr)o(;)262 630 y Ft(where)14 b Fm(\030)g Ft(means)e(that)i(the)g(orbit)f(coun)o(ting)g(sequences)j (\()p Fn(f)1208 636 y Fj(n)1231 630 y Ft(\))e(are)g(the)g(same.)j (\(Unfortu-)262 680 y(nately)m(,)11 b(the)i(homogeneous)e(structure)j (exists)f(only)f(in)f(the)i(case)g(of)f(t)o(w)o(o-graphs.\))17 b(These)262 730 y(relations)c(express)j(formally)11 b(the)j(puzzle)h (at)f(the)g(end)h(of)e(the)i(\014rst)f(section.)324 780 y(It)h(turns)g(out)g(that)g(the)g(sequence)i(\()p Fn(f)926 786 y Fj(n)949 780 y Ft(\()p Fn(H)10 b Ft(W)m(r)d Fn(K)s Ft(\)\))15 b(is)g(not)f(determined)h(b)o(y)g(the)g(cor-)262 829 y(resp)q(onding)h(sequences)i(for)d Fn(H)k Ft(and)d Fn(K)s Ft(.)24 b(W)m(e)15 b(need)i(the)f(sequence)i(\()p Fn(f)1396 835 y Fj(n)1419 829 y Ft(\()p Fn(H)s Ft(\)\))e(and)g(more)262 879 y(detailed)g(information)d(ab)q(out)j Fn(K)s Ft(.)25 b(Later,)16 b(I)h(will)d(describ)q(e)k(what)e(information)d(w)o(e)k (ac-)262 929 y(tually)f(need.)29 b(Here,)18 b(I)f(will)f(describ)q(e)j (the)f(situation)e(in)h(t)o(w)o(o)g(particularly)f(imp)q(ortan)o(t)262 979 y(examples.)h(W)m(e)c(ha)o(v)o(e)471 1113 y Fn(f)491 1119 y Fj(H)c Fl(W)m(r)d Fj(S)600 1113 y Ft(\()p Fn(t)p Ft(\))12 b(=)713 1061 y Fk(1)703 1074 y Fg(Y)703 1162 y Fj(i)p Fl(=1)757 1113 y Ft(\(1)d Fm(\000)g Fn(t)859 1096 y Fj(i)873 1113 y Ft(\))889 1096 y Fk(\000)p Fj(f)931 1100 y Ff(i)945 1096 y Fl(\()p Fj(H)r Fl(\))1014 1113 y Ft(=)i(exp)1128 1030 y Fg(0)1128 1105 y(@)1177 1061 y Fk(1)1164 1074 y Fg(X)1165 1162 y Fj(j)r Fl(=1)1236 1085 y Fn(f)1256 1091 y Fj(H)1288 1085 y Ft(\()p Fn(t)1319 1070 y Fj(j)1336 1085 y Ft(\))f Fm(\000)f Ft(1)p 1236 1104 189 2 v 1320 1142 a Fn(j)1429 1030 y Fg(1)1429 1105 y(A)1472 1113 y Fn(;)262 1248 y Ft(while)764 1307 y Fn(f)784 1313 y Fj(H)g Fl(W)m(r)d Fj(A)896 1307 y Ft(\()p Fn(t)p Ft(\))12 b(=)1079 1278 y(1)p 1004 1297 171 2 v 1004 1335 a(2)d Fm(\000)g Fn(f)1095 1341 y Fj(H)1127 1335 y Ft(\()p Fn(t)p Ft(\))1179 1307 y Fn(:)262 1401 y Ft(These)21 b(relations)f(also)f(describ)q(e)j(the)f(coun)o(ting)f(functions)g(for) g(the)h(comp)q(ositions)d(of)262 1450 y(classes)d(of)e(structures)j (with)e Fn(S)j Ft(or)c Fn(A)p Ft(.)324 1500 y(I)20 b(will)e(tak)o(e)i (the)g(viewp)q(oin)o(t)f(that,)i(with)f(an)o(y)f(oligomorphic)e(group)j Fn(K)s Ft(,)h(there)g(is)262 1550 y(asso)q(ciated)13 b(an)f(op)q(erator)h(\(whic)o(h)f(I)g(also)g(denote)h(b)o(y)g Fn(K)s Ft(\))f(on)h(in)o(teger)f(sequences,)j(so)e(that)726 1637 y(\()p Fn(f)762 1643 y Fj(n)785 1637 y Ft(\()p Fn(H)d Ft(W)m(r)c Fn(K)s Ft(\)\))12 b(=)g Fn(K)s Ft(\()p Fn(f)1108 1643 y Fj(n)1131 1637 y Ft(\()p Fn(H)s Ft(\)\))p Fn(:)262 1724 y Ft(If)h(con)o(v)o(enien)o(t,)g(the)h(op)q(erator)g(can)g(b)q(e)g (tak)o(en)f(to)h(act)f(on)h(generating)f(functions.)18 b(So,)13 b(for)262 1773 y(example,)e(if)g(the)i(sequence)i Fn(f)i Ft(coun)o(ts)d(connected)g(graphs)f(of)f(some)f(t)o(yp)q(e)i (\(e.g.)18 b(Eulerian)262 1823 y(graphs\),)12 b(then)g Fn(S)r(f)17 b Ft(coun)o(ts)c(disjoin)o(t)d(unions)i(of)f(suc)o(h)i (graphs)e(\(e.g.)18 b(ev)o(en)12 b(graphs\),)g(while)262 1873 y Fn(Af)19 b Ft(also)14 b(describ)q(es)i(disjoin)o(t)e(unions)g (but)h(where)g(there)h(is)e(a)h(total)e(order)j(on)e(the)h(set)g(of)262 1923 y(comp)q(onen)o(ts.)h(Bernstein)d(and)e(Sloane)g([3)o(])g(refer)h (to)f(the)h(op)q(erators)g Fn(S)i Ft(and)d Fn(A)g Ft(as)h(EULER)262 1973 y(and)h(INVER)m(T)h(resp)q(ectiv)o(ely)m(.)324 2022 y(There)g(is)f(also)f(a)h Fo(pr)n(o)n(duct)h(action)f Ft(of)f(the)i(wreath)f(pro)q(duct,)h(on)f(the)g(set)h(of)f(functions) 262 2072 y(from)f(\001)i(to)g(\000.)19 b(It)14 b(is)h(not)f(oligomo)o (rphic)e(unless)j Fn(H)i Ft(is)d(oligomorphic)e(and)i Fn(K)j Ft(is)d(a)g(\014nite)262 2122 y(p)q(erm)o(utation)e(group)i (\(that)g(is,)f(\001)h(is)g(\014nite\).)k(As)d(in)e(the)i(case)g(of)e (the)h(direct)h(pro)q(duct,)g(I)262 2172 y(will)d(not)i(consider)h (this)e(action.)262 2308 y Fp(5)66 b(N-free)22 b(graphs)h(and)f(p)r (osets)262 2399 y Ft(In)11 b(an)f(exp)q(erimen)o(t)h(in)o(v)o(olving)e (a)i(n)o(um)o(b)q(er)f(of)g(n)o(uisance)i(factors)f(with)g(discrete)i (lev)o(els,)e(the)262 2449 y(statistician)h(needs)i(to)f(allo)o(w)e (for)h(the)h(fact)g(that)g(eac)o(h)g(n)o(uisance)g(factor)g(ma)o(y)e (con)o(tribute)967 2574 y(8)p eop %%Page: 9 9 9 8 bop 978 721 a Fe(u)907 602 y(u)836 484 y(u)1048 602 y(u)978 484 y(u)907 366 y(u)978 721 y Fd(\024)1002 679 y(\024)1024 644 y(\024)907 602 y(\024)932 561 y(\024)953 526 y(\024)836 484 y(\024)861 443 y(\024)882 408 y(\024)953 721 y(T)928 679 y(T)903 638 y(T)878 596 y(T)853 555 y(T)836 526 y(T)1024 602 y(T)999 561 y(T)974 519 y(T)949 478 y(T)924 436 y(T)907 408 y(T)1084 614 y Ft(sheep)1013 496 y(houses)942 378 y(exp)q(erimen)o(t)694 732 y(sheep-mon)o(ths)623 614 y(house-mon)o(ths)670 496 y(mon)o(ths)752 912 y(Figure)14 b(2:)j(An)e(exp)q(erimen)o(t)895 1189 y Fe(u)895 1071 y(u)p 894 1189 2 119 v 1060 1130 a(u)765 1201 y Ft(sheep)741 1083 y(houses)1096 1142 y(mon)o(ths)816 1381 y(Figure)f(3:)k(A)c(p)q (oset)262 1514 y(to)k(the)g(v)n(ariance)g(of)g(resp)q(onses.)33 b(The)19 b(relationship)f(among)e(these)k(factors)e(therefore)262 1563 y(needs)d(to)e(b)q(e)i(clari\014ed)f(b)q(efore)h(the)f(exp)q (erimen)o(t)g(can)g(b)q(e)g(designed)h(\(that)f(is,)f(b)q(efore)i(the) 262 1613 y(assignmen)o(t)i(of)h(treatmen)o(ts)h(to)f(exp)q(erimen)o (tal)g(units)g(can)h(b)q(e)g(decided\).)34 b(Here)19 b(is)g(an)262 1663 y(example.)f(Supp)q(ose)e(that)f(w)o(e)f(are)i (testing)f(v)n(arious)e(treatmen)o(ts)i(on)g(sheep.)21 b(The)15 b(sheep)262 1713 y(are)j(k)o(ept)h(in)e(a)h(n)o(um)o(b)q(er)f (of)h(houses)h(for)f(a)g(n)o(um)o(b)q(er)f(of)h(mon)o(ths)e(\(a)i(mon)o (th)f(b)q(eing)h(the)262 1763 y(p)q(erio)q(d)f(of)f(one)h(treatmen)o (t\).)26 b(A)17 b(single)f(exp)q(erimen)o(tal)g(unit)h(is)f(a)h(sheep)h (for)e(a)h(mon)o(th,)262 1812 y(or)f(a)f(sheep-mon)o(th.)25 b(The)17 b(relev)n(an)o(t)f(n)o(uisance)g(factors)h(\(apart)f(from)f (trivial)f(ones\))j(are)262 1862 y(houses,)12 b(sheep,)i(house-mon)o (ths,)d(and)h(mon)o(ths,)e(whic)o(h)i(are)h(partially)d(ordered)j(as)g (sho)o(wn)262 1912 y(in)g(Figure)h(2.)324 1962 y(This)g(p)q(oset)h(is)f (a)g(distributiv)o(e)f(lattice,)h(and)g(hence)i(is)e(represen)o(table)i (as)e(the)h(lattice)262 2012 y(of)j(ancestral)j(sets)g(\(up-sets\))g (in)e(a)g(simpler)f(p)q(oset,)k(formed)c(b)o(y)h(sheep,)j(houses,)g (and)262 2061 y(mon)o(ths,)12 b(as)i(in)f(Figure)h(3.)324 2111 y(In)i(statistical)g(terminology)m(,)e(sheep)k(are)e Fo(neste)n(d)h Ft(\(!\))25 b(in)16 b(houses,)i(since)f(there)h(is)e(no) 262 2161 y(relation)f(b)q(et)o(w)o(een)j(the)e(\014fth)g(sheep)i(\(sa)o (y\))e(in)g(di\013eren)o(t)h(houses.)25 b(On)17 b(the)f(other)h(hand,) 262 2211 y(houses)i(and)e(mon)o(ths)g(are)i Fo(cr)n(osse)n(d)p Ft(,)f(since)h(b)q(oth)f(\\same)f(house")i(and)f(\\same)e(mon)o(th")262 2261 y(are)i(p)q(oten)o(tially)g(signi\014can)o(t.)31 b(In)19 b(general,)g Fo(cr)n(ossing)f Ft(t)o(w)o(o)g(p)q(osets)i (consists)g(of)d(taking)262 2311 y(their)e(disjoin)o(t)f(union,)g(and)h Fo(nesting)h Ft(them)e(to)h(taking)f(their)i(ordered)g(sum)e(\(where)i (one)262 2360 y(is)d(ab)q(o)o(v)o(e)h(the)g(other\).)19 b(Statisticians)13 b(had)h(w)o(ork)o(ed)g(out)f(rules)i(for)e(dealing)g (with)g(nesting)262 2410 y(and)g(crossing)h(and)g(their)g(iterates)g ([23)o(],)f(but)h(it)f(turns)i(out)e(that)h(a)f(similar)e(analysis)i (can)967 2574 y(9)p eop %%Page: 10 10 10 9 bop 889 662 a Fe(u)152 b(u)889 366 y(u)g(u)p 888 662 2 296 v 1065 662 V 889 408 a Fd(T)914 449 y(T)939 491 y(T)964 532 y(T)989 574 y(T)1014 615 y(T)1038 657 y(T)1041 662 y(T)871 853 y Ft(Figure)14 b(4:)k(N)262 986 y(b)q(e)e(dev)o(elop)q(ed)h(for)f(n)o(uisance)g(factors)h(based)g (on)e(an)o(y)h(p)q(oset)h(\(a)f Fo(p)n(oset)h(blo)n(ck)g(structur)n(e)p Ft(,)262 1036 y(see)e(Sp)q(eed)g(and)e(Bailey)h([36)o(]\).)324 1086 y(P)o(oset)h(blo)q(c)o(k)f(structures)k(giv)o(e)c(a)g(large)g (class)h(of)f(imprimi)o(tiv)o(e)e(asso)q(ciation)i(sc)o(hemes)262 1135 y(whose)h Fn(P)20 b Ft(and)15 b Fn(Q)g Ft(matrices)g(can)g(b)q(e)g (calculated)g(exactly)m(.)21 b(Moreo)o(v)o(er,)16 b(they)f(are)h(homo-) 262 1185 y(geneous)g(\(assuming)f(the)h(p)q(oset)h(is)f(\014nite;)h (the)f(asso)q(ciation)g(sc)o(heme)g(ma)o(y)e(b)q(e)i(\014nite)h(or)262 1235 y(in\014nite\).)23 b(But)17 b(m)o(y)d(concern)j(here)g(is)f(the)g (question,G(p)q(OED)H(B)O(Y)E(贝利)H((1)O():21 B(HO)O(W)16(B)(T)O(YP)262 262 Y(IIC)E(是)I(结构)H(得到)E(B)O(Y)G(嵌套)H(和)F(交叉)?(o)(f)(262)1335 y(p)q(OsEt)f(())g((g)g)(a)g(w)o(a)o(y)m((f)m(f))(h)(o)(g)(do)q(es)h(this)f(n)O(m)o(b)q(ER)g(比较)f(to)h(h)(262)1385 y(n)O((m)O(b)q(ER)g(f)g(p)q(OsETs))23 b(in)15 b(特),g(Ho)324 1434 y(The)f(sym)o(b)q(ol)e(N)h(will)f(denote)j(the)f(graph)g(or)f(the) h(p)q(oset)h(whic)o(h)e(is)h(sho)o(wn)f(in)h(Figure)f(4.)262 1484 y(A)h(graph)h(or)f(p)q(oset)i(is)e(called)g Fo(N-fr)n(e)n(e)g Ft(if)g(it)g(do)q(esn't)h(con)o(tain)f(N)h(as)f(an)h(induced)g (substruc-)262 1534 y(ture.)18 b(The)11 b(class)h(of)f(N-free)h(graphs) f(has)h(b)q(een)g(studied)g(in)f(man)o(y)e(con)o(texts,)j(under)g(man)o (y)262 1584 y(di\013eren)o(t)i(names.)k(I)13 b(summarise)f(the)j(main)d (facts.)324 1667 y Fm(\017)20 b Ft(The)15 b(complemen)o(t)c(of)j(an)f (N-free)i(graph)f(is)f(N-free.)324 1750 y Fm(\017)20 b Ft(An)14 b(N-free)g(graph)f(with)g(more)g(than)g(one)h(v)o(ertex)g (is)f(connected)j(if)c(and)h(only)g(if)g(its)365 1800 y(complemen)o(t)f(is)i(disconnected.)324 1883 y Fm(\017)20 b Ft(The)15 b(class)g(of)f(N-free)i(graphs)f(is)f(the)h(smallest)f (class)h(con)o(taining)e(the)j(one-v)o(ertex)365 1932 y(graph)e(and)g(closed)g(under)h(complemen)o(tation)c(and)j(disjoin)o (t)e(union.)324 2015 y Fm(\017)20 b Ft(The)15 b(edges)g(of)e(an)g (N-free)i(graph)f(can)g(b)q(e)g(orien)o(ted)h(to)f(form)e(an)h(N-free)i (p)q(oset.)324 2098 y Fm(\017)20 b Ft(A)14 b(p)q(oset)g(is)f(N-free)h (if)e(and)i(only)e(if)h(it)f(can)i(b)q(e)g(built)e(from)g(the)i (one-elemen)o(t)f(p)q(oset)365 2148 y(b)o(y)h(nesting)g(and)g (crossing.)324 2231 y(W)m(e)h(see)i(that,)e(for)g Fn(n)f(>)h Ft(1,)g(the)h(n)o(um)o(b)q(ers)f(of)g(connected)i(and)f(disconnected)h (N-free)262 2281 y(graphs)f(on)f Fn(n)h Ft(v)o(ertices)h(are)f(equal.) 24 b(Let)16 b Fn(a)g Ft(b)q(e)h(the)f(sequence)i(en)o(umerating)d (connected)262 2331 y(N-free)f(graphs.)k(Then)d(w)o(e)f(ha)o(v)o(e)640 2422 y Fn(a)662 2428 y Fl(1)692 2422 y Ft(=)e(1)p Fn(;)89 b Ft(\()p Fn(S)r(a)p Ft(\))939 2428 y Fj(n)974 2422 y Ft(=)12 b(2)p Fn(a)1061 2428 y Fj(n)1139 2422 y Ft(for)h Fn(n)f(>)g Ft(1)p Fn(:)957 2574 y Ft(10)p eop %%Page: 11 11 11 10 bop 262 307 a Ft(This)16 b(giv)o(es)h(a)g(recurrence)j(relation)c (for)h Fn(a)950 313 y Fj(n)973 307 y Ft(,)g(since)h(\()p Fn(S)r(a)p Ft(\))1188 313 y Fj(n)1228 307 y Ft(is)f(equal)g(to)f Fn(a)1461 313 y Fj(n)1501 307 y Ft(plus)h(terms)262 357 y(in)o(v)o(olving)c Fn(a)463 363 y Fj(i)493 357 y Ft(for)i Fn(i)g(<)h(n)p Ft(;)g(so)g(the)g(n)o(um)o(b)q(ers)g(are)g(easily)g (calculated.)24 b(It)16 b(is)g(not)f(an)h(easy)262 407 y(recurrence)g(to)d(solv)o(e,)f(but)i(it)e(can)i(b)q(e)f(sho)o(wn)g (that)g(the)h(sequence)h(gro)o(ws)e(exp)q(onen)o(tially)m(.)262 457 y(The)h(n)o(um)o(b)q(er)f Fn(a)520 463 y Fj(n)556 457 y Ft(is)h(a)g(lo)o(w)o(er)f(b)q(ound)h(for)g(the)g(n)o(um)o(b)q(er) f(of)h(N-free)g(p)q(osets.)324 506 y(W)m(e)f(\\brac)o(k)o(et")h(the)g (n)o(um)o(b)q(er)e(of)h(N-free)i(p)q(osets)f(as)g(follo)o(ws.)i(An)e Fo(N-fr)n(e)n(e)f(bip)n(oset)h Ft(is)f(a)262 556 y(set)g(supp)q(orting) g(t)o(w)o(o)g(p)q(osets,)h(whic)o(h)e(are)i(complemen)o(tary)c(\(in)j (the)g(sense)i(that)e(an)o(y)f(t)o(w)o(o)262 606 y(distinct)17 b(p)q(oin)o(ts)g(are)g(comparable)e(in)i(exactly)g(one)g(of)g(the)g(p)q (osets\))h(and)f(b)q(oth)g(N-free.)262 656 y(An)o(y)11 b(N-free)g(graph)g(and)g(its)h(complemen)o(t)d(can)i(b)q(e)h(orien)o (ted)f(to)g(form)f(an)h(N-free)h(bip)q(oset.)262 706 y(\()p Fo(Exer)n(cise:)22 b Ft(sho)o(w)16 b(that,)h(if)e(w)o(e)h(set)h Fn(x)e(<)h(y)i Ft(when)e(this)h(relation)e(holds)h(in)g(either)h(p)q (oset)262 756 y(of)d(an)h(N-free)h(bip)q(oset,)f(the)h(result)g(is)f(a) f(total)h(order.\))22 b(Giv)o(en)15 b(the)g(order)h(1)e Fn(<)f Fm(\001)7 b(\001)g(\001)12 b Fn(<)i(m)262 805 y Ft(and)f(bip)q(osets)i Fn(B)534 811 y Fl(1)553 805 y Ft(,)e(.)7 b(.)f(.)h(,)13 b Fn(B)690 811 y Fj(m)722 805 y Ft(,)h(w)o(e)g(can)g(com)o(bine)e(them)h(to)h(get)g(a)g(new)g (bip)q(oset)h Fn(B)h Ft(whose)262 855 y(diconnected)h(p)q(oset)h(is)e (the)h(disjoin)o(t)f(union)g(of)g(the)h(connected)h(p)q(osets)g(of)e (the)h Fn(B)1596 861 y Fj(i)1627 855 y Ft(and)262 905 y(whose)c(connected)i(p)q(oset)f(is)f(the)h(ordered)g(sum)e(of)h(the)g (disconnected)i(p)q(osets)g(of)d(the)i Fn(B)1668 911 y Fj(i)1682 905 y Ft(.)262 955 y(Hence,)f(if)f Fn(b)g Ft(is)h(the)g(sequence)i(en)o(umerating)c(N-free)i(bip)q(osets)h(for)e (whic)o(h)g(the)h(\014rst)h(p)q(oset)262 1005 y(is)f(connected,)j(then) e(the)h(total)e(n)o(um)o(b)q(er)g(of)h(N-free)g(bip)q(osets)h(is)f(2)p Fn(b)1339 1011 y Fj(n)1375 1005 y Ft(for)g Fn(n)d(>)h Ft(1,)i(and)f(w)o(e)262 1054 y(ha)o(v)o(e)645 1104 y Fn(b)663 1110 y Fl(1)693 1104 y Ft(=)e(1)p Fn(;)90 b Ft(\()p Fn(Ab)p Ft(\))940 1110 y Fj(n)974 1104 y Ft(=)12 b(2)p Fn(b)1057 1110 y Fj(n)1134 1104 y Ft(for)i Fn(n)d(>)h Ft(1)p Fn(:)262 1169 y Ft(This)17 b(also)g(giv)o(es)g(a)g(recurrence)k (whic)o(h)c(implies)f(that)h Fn(b)1173 1175 y Fj(n)1213 1169 y Ft(gro)o(ws)g(exp)q(onen)o(tially)m(.)28 b(This)262 1219 y(recurrence)14 b(can)e(b)q(e)g(solv)o(ed)g(explicitly:)k(if)11 b Fn(b)p Ft(\()p Fn(t)p Ft(\))g(is)h(the)g(generating)g(function,)f (and)h Fn(u)p Ft(\()p Fn(t)p Ft(\))f(=)262 1269 y Fn(b)p Ft(\()p Fn(t)p Ft(\))e Fm(\000)g Ft(1)14 b(\(so)g(that)g Fn(u)p Ft(\(0\))d(=)h(0\),)h(w)o(e)i(ha)o(v)o(e)769 1345 y(1)p Fn(=)p Ft(\(1)8 b Fm(\000)i Fn(u)p Ft(\))h(=)h(1)d(+)h(2)p Fn(u)e Fm(\000)i Fn(t;)262 1420 y Ft(giving)g Fn(u)i Ft(=)468 1404 y Fl(1)p 468 1411 17 2 v 468 1434 a(4)489 1420 y Ft(\(1)6 b(+)g Fn(t)g Fm(\000)629 1386 y(p)p 666 1386 192 2 v 666 1420 a Ft(1)j Fm(\000)g Ft(6)p Fn(t)g Ft(+)h Fn(t)839 1408 y Fl(2)857 1420 y Ft(\).)18 b(The)13 b(Binomial)d(Theorem)i(no)o(w)g(giv)o(es)g(a)g(form)o(ula)262 1476 y(for)i(the)h(co)q(e\016cien)o(ts.)22 b(The)15 b(function)g Fn(u)f Ft(has)h(a)g(singularit)o(y)e(at)i Fn(t)e Ft(=)g(3)c Fm(\000)i Ft(2)1451 1442 y Fm(p)p 1485 1442 21 2 v 1485 1476 a Ft(2,)j(so)h(this)g(is)262 1526 y(its)e(radius)h(of)g(con)o(v)o (ergence,)h(and)e(the)i(exp)q(onen)o(tial)e(constan)o(t)i(is)f(3)9 b(+)g(2)1409 1492 y Fm(p)p 1443 1492 V 1443 1526 a Ft(2.)324 1576 y(No)o(w)14 b(let)h Fn(c)f Ft(and)g Fn(d)g Ft(b)q(e)i(the)f (sequences)i(en)o(umerating)c(connected)j(and)f(disconnected)262 1626 y(N-free)k(p)q(osets,)h(where)f(w)o(e)g(use)g(the)g(strange)g(con) o(v)o(en)o(tion)f(that)g Fn(c)1350 1632 y Fl(1)1387 1626 y Ft(=)i Fn(d)1461 1632 y Fl(1)1498 1626 y Ft(=)f(1.)31 b(This)262 1676 y(case)16 b(is)g(a)f(curious)i(mixture)d(of)h(the)i(t)o (w)o(o)e(preceding.)25 b(Since)16 b(an)o(y)f(disconnected)j(N-free)262 1725 y(p)q(oset)c(is)f(a)f(disjoin)o(t)h(union)f(of)h(connected)i (ones,)e(and)g(an)o(y)g(connected)i(N-free)f(p)q(oset)g(\(on)262 1775 y(more)c(than)h(one)g(elemen)o(t\))g(an)g(ordered)h(sum)e(of)h (disconnected)h(ones,)g(w)o(e)g(get)f(the)h(m)o(utual)262 1825 y(recurrence)480 1900 y Fn(c)498 1906 y Fl(1)528 1900 y Ft(=)g Fn(d)594 1906 y Fl(1)623 1900 y Ft(=)g(1)p Fn(;)89 b Ft(\()p Fn(S)r(c)p Ft(\))866 1906 y Fj(n)902 1900 y Ft(=)12 b(\()p Fn(Ad)p Ft(\))1031 1906 y Fj(n)1065 1900 y Ft(=)f Fn(c)1126 1906 y Fj(n)1158 1900 y Ft(+)f Fn(d)1222 1906 y Fj(n)1299 1900 y Ft(for)k Fn(n)d(>)h Ft(1)p Fn(:)324 1976 y Ft(This)h(enables)i(the)f(sequences)i(to)e(b)q (e)g(calculated.)k(They)c(gro)o(w)f(exp)q(onen)o(tially)m(,)f(with)262 2026 y(exp)q(onen)o(tial)19 b(constan)o(t)h(appro)o(ximately)d(4.62)i (\(see)i(Cameron)d([10)o(])h(for)g(more)g(precise)262 2076 y(asymptotics\).)30 b(If)18 b Fn(c)p Ft(\()p Fn(t)p Ft(\))h(and)f Fn(d)p Ft(\()p Fn(t)p Ft(\))g(are)h(the)g(generating)f (functions)h(of)f(the)h(sequences,)262 2125 y(then)559 2198 y Fn(c)p Ft(\()p Fn(t)p Ft(\))10 b(+)f Fn(d)p Ft(\()p Fn(t)p Ft(\))g Fm(\000)h Fn(t)f Fm(\000)g Ft(1)j(=)1001 2170 y(1)p 941 2189 141 2 v 941 2227 a(2)d Fm(\000)h Fn(d)p Ft(\()p Fn(t)p Ft(\))1098 2198 y(=)1153 2146 y Fk(1)1143 2159 y Fg(Y)1142 2247 y Fj(i)p Fl(=1)1196 2198 y Ft(\(1)f Fm(\000)h Fn(t)1299 2181 y Fj(i)1312 2198 y Ft(\))1328 2181 y Fk(\000)p Fj(c)1369 2185 y Ff(i)1385 2198 y Fn(:)324 2300 y Ft(In)g(an)o(y)g(case,)i(w)o(e)f(ha)o(v)o(e)f (more)g(than)g(enough)h(information)c(to)k(answ)o(er)g(the)g(motiv)n (ating)262 2350 y(question.)29 b(Since)19 b(there)g(are)f(roughly)f(2) 926 2334 y Fj(n)947 2322 y Fc(2)962 2334 y Fj(=)p Fl(4)1016 2350 y Ft(p)q(osets)i(altogether)f(\(indeed,)h(this)e(man)o(y)262 2399 y(t)o(w)o(o-lev)o(el)c(p)q(osets\),)j(only)e(a)h(v)n(anishingly)e (small)f(prop)q(ortion)j(of)f(them)g(are)h(obtained)g(b)o(y)262 2449 y(nesting)f(and)f(crossing.)957 2574 y(11)p eop %%Page: 12 12 12 11 bop 262 307 a Fp(6)66 b(Algebraic)24 b(in)n(terlude)262 398 y Ft(There)14 b(is)f(a)g(graded)h(algebra)f(whic)o(h)g(can)g(b)q(e) h(constructed)i(from)11 b(a)i(p)q(erm)o(utation)f(group,)262 448 y(suc)o(h)17 b(that)g(the)h(dimensions)e(of)g(its)h(homogeneous)g (comp)q(onen)o(ts)f(are)i(the)f(n)o(um)o(b)q(ers)g(of)262 498 y(orbits)12 b(of)g(the)i(group)e(on)g Fn(n)p Ft(-sets.)19 b(Its)13 b(algebraic)f(structure)j(can)e(giv)o(e)f(a)g(bit)h(more)e (insigh)o(t)262 548 y(in)o(to)i(the)h(com)o(binatorics)f(of)g(the)h (orbits.)324 597 y(F)m(or)d(an)o(y)h(in\014nite)g(set)g(\012,)g(let)g Fn(V)810 603 y Fj(n)845 597 y Ft(denote)h(the)f(set)h(of)e(all)g (functions)h(from)1483 564 y Fg(\000)1502 579 y Fl(\012)1504 612 y Fj(n)1526 564 y Fg(\001)1557 597 y Ft(\(the)h(set)262 647 y(of)f Fn(n)p Ft(-elemen)o(t)h(subsets)i(of)e(\012\))g(to)g(y)o (our)h(fa)o(v)o(ourite)e(\014eld)i(of)e(c)o(haracteristic)j(zero)f (\(whic)o(h)g(I)262 697 y(will)f(tak)o(e)h(to)h(b)q(e)g(the)g(rational) f(n)o(um)o(b)q(ers)g(here\).)22 b(Eac)o(h)15 b Fn(V)1189 703 y Fj(n)1226 697 y Ft(is)g(a)f(rational)g(v)o(ector)h(space,)262 747 y(and)e Fn(V)366 753 y Fl(0)399 747 y Ft(has)h(dimension)e(1)i (\(there)h(is)f(only)f(one)h(empt)o(y)f(set\).)19 b(No)o(w)14 b(let)875 868 y Fm(A)e Ft(=)978 816 y Fk(1)964 829 y Fg(M)964 917 y Fj(n)p Fl(=0)1033 868 y Fn(V)1057 874 y Fj(n)262 992 y Ft(b)q(e)i(the)h(direct)g(sum)f(of)f(these)j(spaces.)k (W)m(e)14 b(de\014ne)h(a)f(m)o(ultiplication)d(on)j Fm(A)g Ft(b)o(y)h(the)f(rule)262 1042 y(that,)f(for)h(an)o(y)f Fn(f)j Fm(2)11 b Fn(V)604 1048 y Fj(k)625 1042 y Ft(,)i Fn(g)g Fm(2)e Fn(V)746 1048 y Fj(l)759 1042 y Ft(,)j(the)g(pro)q(duct)h Fn(f)t(g)h Ft(is)e(the)g(function)g(in)f Fn(V)1420 1048 y Fj(k)q Fl(+)p Fj(l)1491 1042 y Ft(de\014ned)i(b)o(y)693 1138 y Fn(f)t(g)q Ft(\()p Fn(M)5 b Ft(\))12 b(=)898 1098 y Fg(X)871 1195 y Fj(K)r Fk(2)923 1198 y Ft(\()940 1182 y Ff(M)946 1205 y(k)968 1198 y Ft(\))991 1138 y Fn(f)t Ft(\()p Fn(K)s Ft(\))p Fn(g)q Ft(\()p Fn(M)k Fm(n)9 b Fn(K)s Ft(\))262 1287 y(for)i(an)o(y)g(\()p Fn(k)6 b Ft(+)f Fn(l)q Ft(\)-set)14 b Fn(M)5 b Ft(.)17 b(This)11 b(mak)o(es)g Fm(A)h Ft(a)f(comm)o(utativ)o(e,)e(asso)q(ciativ)o(e,)j (graded)g(algebra)262 1336 y(o)o(v)o(er)g Fi(Q)p Ft(.)j(\(It)e(is)g(in) f(fact)h(the)h(reduced)g(incidence)g(algebra)e(of)g(the)i(p)q(oset)g (of)e(\014nite)h(subsets)262 1386 y(of)d(\012,)g(but)h(this)g(fact)g (pla)o(ys)f(no)g(role)h(here.)18 b(I)11 b(also)f(remark)g(that)g(Glynn) g([14)o(])g(has)h(made)f(use)262 1436 y(of)i(a)h(similar)e(algebra,)i (where)h(the)g(supp)q(orts)h(of)e(the)h Fn(k)q Ft(-set)g(and)f Fn(l)q Ft(-set)h(to)g(whic)o(h)f Fn(f)18 b Ft(and)13 b Fn(g)262 1486 y Ft(are)f(applied)g(in)g(de\014ning)g(the)h(pro)q (duct)h(are)e(not)h(required)g(to)f(b)q(e)h(disjoin)o(t.)k(This)12 b(algebra)262 1536 y(has)i(v)o(ery)g(di\013eren)o(t)h(prop)q(erties.)k (Glynn)13 b(uses)i(it)f(to)f(study)i(reconstruction)g(problems.\))324 1585 y(An)d(elemen)o(t)g(of)g Fn(V)612 1591 y Fj(n)648 1585 y Ft(is)g(called)g(a)g Fo(homo)n(gene)n(ous)j(element)f(of)f(de)n (gr)n(e)n(e)h Fn(n)e Ft(in)g(the)h(algebra)262 1635 y Fm(A)p Ft(.)k(\(This)12 b(has)g(no)g(connection)h(with)f(our)g(earlier) g(usage)g(of)g(the)g(w)o(ord)g(\\homogeneous".\))262 1685 y(A)j(particular)g(homogeneous)g(elemen)o(t)g(of)g(degree)i(1)e (is)h(the)g(constan)o(t)g(function)f Fn(e)h Ft(with)262 1735 y(v)n(alue)e(1.)20 b(Multiplication)13 b(b)o(y)i Fn(e)g Ft(induces)h(a)f(linear)f(map)f(from)g Fn(V)1305 1741 y Fj(n)1343 1735 y Ft(to)i Fn(V)1419 1741 y Fj(n)p Fl(+1)1498 1735 y Ft(for)g(eac)o(h)g Fn(n)p Ft(;)262 1785 y(this)e(map)f(is)i(represen)o(ted)i(b)o(y)e(the)g(matrix)e Fn(P)19 b Ft(of)13 b(Section)h(2,)f(and)h(Theorem)f(3.1)g(implies)262 1835 y(that)g(it)h(is)g(a)f(non-zero-divisor.)324 1884 y(No)o(w)f(let)h Fn(G)g Ft(b)q(e)h(a)e(p)q(erm)o(utation)g(group)h(on)g (\012.)k(Then)d Fn(G)e Ft(acts)i(on)f(eac)o(h)g(space)h Fn(V)1602 1890 y Fj(n)1625 1884 y Ft(,)f(b)o(y)262 1934 y(p)q(erm)o(uting)f(the)i(argumen)o(ts)f(of)f(the)i(functions.)k(Let)c Fn(V)1155 1919 y Fj(G)1145 1944 y(n)1196 1934 y Ft(b)q(e)g(the)g(space) h(of)e(functions)g(in)262 1984 y Fn(V)286 1990 y Fj(n)321 1984 y Ft(\014xed)h(b)o(y)f Fn(G)p Ft(.)k(Since)d(a)e(function)h(is)g (\014xed)h(b)o(y)f Fn(G)f Ft(if)g(and)h(only)g(if)f(it)h(is)g(constan)o (t)g(on)g(the)262 2034 y(orbits)g(of)h Fn(G)p Ft(,)f(w)o(e)h(ha)o(v)o (e)815 2084 y(dim)n(\()p Fn(V)933 2066 y Fj(G)924 2094 y(n)961 2084 y Ft(\))e(=)g Fn(f)1053 2090 y Fj(n)1076 2084 y Ft(\()p Fn(G)p Ft(\))262 2156 y(if)h Fn(G)g Ft(is)h(oligomo)o (rphic.)i(F)m(urthermore,)d(w)o(e)h(de\014ne)854 2275 y Fm(A)887 2258 y Fj(G)927 2275 y Ft(=)985 2224 y Fk(1)972 2236 y Fg(X)970 2324 y Fj(n)p Fl(=0)1040 2275 y Fn(V)1073 2258 y Fj(G)1064 2286 y(n)262 2399 y Ft(to)h(b)q(e)i(the)f(set)h(of)e (\014xed)h(p)q(oin)o(ts)g(of)f Fn(G)h Ft(in)f Fm(A)p Ft(.)24 b(If)16 b Fn(G)f Ft(\014xes)i(t)o(w)o(o)e(functions,)h(it)g (\014xes)g(their)262 2449 y(pro)q(duct;)g(so)g Fm(A)517 2434 y Fj(G)560 2449 y Ft(is)g(a)f(subalgebra)g(of)g Fm(A)p Ft(.)23 b(F)m(or)15 b(oligomorphic)d(groups)k Fn(G)p Ft(,)f(w)o(e)h(see)h(that)957 2574 y(12)p eop %%Page: 13 13 13 12 bop 262 307 a Ft(the)13 b(generating)h(function)e Fn(f)716 313 y Fj(G)745 307 y Ft(\()p Fn(t)p Ft(\))h(is)g(the)h(P)o (oincar)o(\023)-20 b(e)13 b(series)i(of)d Fm(A)1276 292 y Fj(G)1304 307 y Ft(.)18 b(In)13 b(particular,)g(if)f Fn(S)k Ft(is)262 357 y(the)g(symmetric)e(group)h(on)g(\012,)h(then)g Fm(A)904 342 y Fj(S)944 357 y Ft(is)f(the)h(p)q(olynomial)d(algebra)i (in)g(one)h(v)n(ariable)262 407 y(o)o(v)o(er)d Fi(Q)p Ft(,)e(the)k(generator)f(b)q(eing)g(the)h(elemen)o(t)e Fn(e)h Ft(de\014ned)h(ab)q(o)o(v)o(e.)324 457 y(If)k Fn(G)f Ft(is)h(oligomorphic,)e(then)j Fn(V)867 442 y Fj(G)858 467 y(n)914 457 y Ft(is)f(spanned)h(b)o(y)f(the)h(c)o (haracteristic)h(functions)262 506 y(of)d(the)h Fn(G)p Ft(-orbits)f(on)g Fn(n)p Ft(-sets;)k(eac)o(h)d(orbit)f(corresp)q(onds)j (to)d(an)h(isomorphism)c(t)o(yp)q(e)k(of)262 556 y Fn(n)p Ft(-elemen)o(t)e(structures)k(in)d(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)18 b(class)h(of)f Fn(G)p Ft(.)30 b(According)19 b(to)f(our)h(philosoph)o(y)m(,)262 606 y(it)e(is)g(p)q(ossible)g(to)h(de\014ne)g(an)f(analogous)f(algebra)h (for)g(more)f(general)i(classes)h(of)d(\014nite)262 656 y(structures.)29 b(I)17 b(lea)o(v)o(e)g(it)f(as)h(an)g(exercise)i(to)e (write)g(out)g(the)g(precise)i(de\014nition)d(of)h(this)262 706 y(algebra.)324 756 y(W)m(e)12 b(no)o(w)h(consider)h(the)f (structure)i(of)e Fm(A)968 740 y Fj(G)1009 756 y Ft(when)g Fn(G)g Ft(is)f(a)h(direct)h(or)f(wreath)g(pro)q(duct.)262 805 y(The)h(direct)h(pro)q(duct)f(is)g(straigh)o(tforw)o(ard:)k(w)o(e)c (ha)o(v)o(e)781 897 y Fm(A)814 879 y Fj(H)r Fk(\002)p Fj(K)913 897 y Ft(=)e Fm(A)990 879 y Fj(H)1031 897 y Fm(\012)1063 903 y Fb(Q)1097 897 y Fm(A)1130 879 y Fj(K)1162 897 y Fn(:)324 988 y Ft(W)m(reath)18 b(pro)q(ducts)j(are)e(more)f (di\016cult,)h(but)g(there)h(are)f(results)h(in)e(some)g(sp)q(ecial)262 1038 y(cases.)h(First,)12 b(let)h Fn(G)e Ft(=)h Fn(S)e Ft(W)m(r)c Fn(K)s Ft(.)18 b(If)13 b Fn(K)j Ft(is)c(a)h(\014nite)g(p)q (erm)o(utation)e(group)i(on)f(a)h(set)h(of)e(size)262 1088 y Fn(n)p Ft(,)i(then)h(it)g(can)g(b)q(e)g(represen)o(ted)j(as)c(a) h(group)f(of)h Fn(n)9 b Fm(\002)h Fn(n)15 b Ft(matrices)f(\(using)h(p)q (erm)o(utation)262 1137 y(matrices)i(corresp)q(onding)i(to)f(the)g (elemen)o(ts)g(of)f Fn(K)s Ft(\).)31 b(Suc)o(h)18 b(a)g(linear)f(group) h Fn(K)j Ft(has)d(a)262 1187 y(ring)d Fn(I)s Ft(\()p Fn(K)s Ft(\))i(of)f(in)o(v)n(arian)o(ts,)f(the)h(p)q(olynomial)d (functions)j(on)g Fi(Q)1278 1172 y Fj(n)1314 1187 y Ft(\014xed)h(b)o(y) f Fn(K)s Ft(.)25 b(It)16 b(turns)262 1237 y(out)e(that)h Fm(A)460 1222 y Fj(S)7 b Fl(W)m(r)g Fj(K)583 1237 y Ft(is)15 b(isomorphic)e(to)h Fn(I)s Ft(\()p Fn(K)s Ft(\).)21 b(In)15 b(particular,)f(the)h(generating)g(function)262 1287 y Fn(f)282 1293 y Fj(S)7 b Fl(W)m(r)g Fj(K)391 1287 y Ft(\()p Fn(t)p Ft(\))13 b(is)f(the)h Fo(Molien)h(series)d Ft([22)o(])h(of)g(the)h(linear)f(group)h Fn(K)s Ft(.)k(If)12 b Fn(K)k Ft(is)c(the)h(symmetric)262 1337 y(group)h Fn(S)405 1343 y Fj(n)442 1337 y Ft(then,)h(b)o(y)f(Newton's)g(Theorem,)g Fn(I)s Ft(\()p Fn(K)s Ft(\))h(is)f(a)g(p)q(olynomial)d(ring)j (generated)i(b)o(y)262 1386 y(the)11 b(elemen)o(tary)f(symmetric)f (functions,)i(whic)o(h)f(ha)o(v)o(e)h(degrees)h(1)p Fn(;)7 b Ft(2)p Fn(;)g(:)g(:)g(:)t(;)g(n)p Ft(;)j(and)h(w)o(e)g(ha)o(v)o(e)735 1509 y Fn(f)755 1515 y Fj(S)d Fl(W)m(r)e Fj(S)852 1519 y Ff(n)875 1509 y Ft(\()p Fn(t)p Ft(\))12 b(=)994 1457 y Fj(n)978 1469 y Fg(Y)978 1558 y Fj(i)p Fl(=1)1032 1509 y Ft(\(1)d Fm(\000)g Fn(t)1134 1492 y Fj(i)1148 1509 y Ft(\))1164 1492 y Fk(\000)p Fl(1)1209 1509 y Fn(:)324 1642 y Ft(There)k(is)g(a)f(completely)f(di\013eren)o(t)i(situation)f (in)g(whic)o(h)g(w)o(e)h(can)f(guaran)o(tee)h(that)g Fm(A)1666 1627 y Fj(G)262 1692 y Ft(is)j(a)h(p)q(olynomial)d(ring)j (generated)h(b)o(y)f(homogeneous)f(elemen)o(ts.)28 b(Supp)q(ose)18 b(that)f Fn(G)g Ft(is)262 1742 y(the)g(automorphism)c(group)j(of)g(a)g (homogeneous)f(structure,)k(whose)e(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)17 b(class)f(has)262 1792 y(a)i(\\go)q(o)q(d)g (notion)f(of)h(connectedness".)35 b(\(I)18 b(will)f(not)i(de\014ne)g (this)g(precisely)m(.)32 b(It)19 b(holds)262 1842 y(for)g(graphs,)h (etc.)36 b(In)20 b(general,)g(what)g(is)f(required)h(is)g(that)f(ev)o (ery)i(structure)g(can)f(b)q(e)262 1892 y(uniquely)11 b(expressed)j(as)e(the)h(disjoin)o(t)e(union)g(of)h(connected)i (structures,)g(and)e(that)g(giv)o(en)262 1941 y(an)g(arbitrary)g (structure)j(and)d(a)g(partition)g(of)f(its)i(p)q(oin)o(ts,)f(the)h (structure)i(\\con)o(tains")c(\(as)262 1991 y(a)17 b(substructure\))j (the)e(disjoin)o(t)f(union)g(of)g(the)h(induced)g(substructures)j(on)c (its)g(parts.\))262 2041 y(Then)f(it)g(can)h(b)q(e)g(sho)o(wn)f(that)h Fm(A)808 2026 y Fj(G)852 2041 y Ft(is)g(a)f(p)q(olynomial)d(algebra.)25 b(Its)17 b(generators)g(are)g(in)262 2091 y(one-to-one)c(corresp)q (ondence)k(with)d(the)g(connected)i(structures.)324 2141 y(No)o(w)e(another)g(in)o(terpretation)h(of)e(the)i Fn(S)r Ft(-transform)f(is)g(that,)g(if)f(a)h(sequence)i Fn(f)j Ft(en)o(u-)262 2190 y(merates)c(the)i(n)o(um)o(b)q(er)e(of)h(p)q (olynomial)c(generators)18 b(of)d(giv)o(en)g(degree)j(in)d(a)h(p)q (olynomial)262 2240 y(algebra,)11 b(then)h(the)g Fn(n)p Ft(th)g(term)f(of)g Fn(S)r(f)17 b Ft(is)12 b(the)g(degree)h(of)e(the)i Fn(n)p Ft(th)e(homogeneous)g(comp)q(on-)262 2290 y(en)o(t)j(of)g(the)h (algebra.)20 b(So)14 b(the)h(relation)f(b)q(et)o(w)o(een)i(connected)g (and)f(arbitrary)f(structures)262 2340 y(is)f(exactly)h(mirrored)f(in)h (the)g(algebra.)324 2390 y(A)h(sp)q(ecial)h(case)g(o)q(ccurs)h(for)e (the)h(group)f Fn(H)10 b Ft(W)m(r)c Fn(S)r Ft(.)24 b(Recall)14 b(that)i(a)f(structure)i(in)e(the)262 2439 y(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)14 b(class)h(of)f(this)g(group)g (consists)h(of)f(a)g(set)h(with)f(a)g(partition,)f(ha)o(ving)g(a)h (structure)957 2574 y(13)p eop %%Page: 14 14 14 13 bop 262 307 a Ft(in)15 b(the)i(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)17 b(class)g(of)f Fn(H)j Ft(on)d(eac)o(h)h(part)f (of)g(the)h(partition.)25 b(T)m(aking)15 b(the)i(connected)262 357 y(structures)12 b(as)e(those)g(with)g(just)g(one)g(part,)g(w)o(e)g (ha)o(v)o(e)f(a)h(\\go)q(o)q(d)f(notion)g(of)g(connectedness";)262 407 y(so)14 b Fm(A)346 392 y Fj(H)8 b Fl(W)m(r)f Fj(S)469 407 y Ft(is)15 b(a)f(p)q(olynomial)d(algebra)j(with)g Fn(f)1023 413 y Fj(n)1046 407 y Ft(\()p Fn(H)s Ft(\))h(generators)h(of) e(degree)h Fn(n)g Ft(for)f(eac)o(h)262 457 y Fn(n)p Ft(.)j(Note)12 b(that)g(the)g(structure)i(of)d Fm(A)825 442 y Fj(H)d Fl(W)m(r)e Fj(S)945 457 y Ft(do)q(es)13 b(not)e(dep)q(end)i(on)f(the)g (detailed)f(structure)262 506 y(of)i Fm(A)342 491 y Fj(H)373 506 y Ft(,)h(only)f(on)h(its)f(P)o(oincar)o(\023)-20 b(e)15 b(series.)324 556 y(I)i(end)i(this)f(section)g(with)g(a)f (puzzle.)31 b(There)19 b(is)e(a)h(coun)o(table)f(homogeneous)g(t)o(w)o (o-)262 606 y(graph,)c(since)i(\014nite)f(t)o(w)o(o-graphs)g(form)e(a)i (F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class.)k(Let)14 b Fn(G)g Ft(b)q(e)h(its)f(automorphism)262 656 y(group,)f(and)g (consider)h Fm(A)665 641 y Fj(G)693 656 y Ft(.)k(Is)c(it)f(a)h(p)q (olynomial)c(algebra?)17 b(The)d(answ)o(er)h(is)e(not)h(kno)o(wn.)262 706 y(If)i(it)g(is,)h(then)g(the)g(n)o(um)o(b)q(er)f(of)g(p)q (olynomial)d(generators)18 b(of)e(degree)i Fn(n)e Ft(is)g(equal)h(to)f (the)262 756 y(n)o(um)o(b)q(er)e(of)g(Eulerian)h(graphs)g(on)g Fn(n)f Ft(v)o(ertices.)23 b(Also,)14 b(ho)o(w)h(do)f(reduced)j(t)o(w)o (o-graphs)d(\014t)262 805 y(in)o(to)f(the)h(picture?)324 855 y(The)21 b(general)f(pattern)i(of)e(this)g(puzzle)i(is)e(a)g(group) h Fn(G)f Ft(for)g(whic)o(h)g(the)h(sequence)262 905 y(\()p Fn(a)300 911 y Fj(n)322 905 y Ft(\))12 b(=)g Fn(S)421 890 y Fk(\000)p Fl(1)466 905 y Ft(\()p Fn(f)502 911 y Fj(n)525 905 y Ft(\()p Fn(G)p Ft(\)\))f(has)g(a)g(natural)f(com)o (binatorial)e(in)o(terpretation;)k(w)o(e)f(w)o(an)o(t)f(to)h(kno)o(w) 262 955 y(whether)k Fm(A)454 940 y Fj(G)496 955 y Ft(is)e(a)h(p)q (olynomial)c(algebra)k(with)f(generators)j(en)o(umerated)d(b)o(y)h(\()p Fn(a)1548 961 y Fj(n)1571 955 y Ft(\).)324 1005 y(Here)e(is)e(an)g (example)g(where)h(this)g(approac)o(h)g(succeeded,)i(and)d(connected)j (the)e(theory)262 1054 y(here)16 b(with)f(a)f(v)o(ery)i(di\013eren)o(t) g(part)f(of)g(mathematics.)k(Let)d Fn(q)g Ft(b)q(e)g(a)f(p)q(ositiv)o (e)f(in)o(teger.)23 b(It)262 1104 y(is)14 b(kno)o(wn)g(that)g(there)i (is)e(a)h(partition)e(of)h(the)h(set)g(of)f(rational)f(n)o(um)o(b)q (ers)i(in)o(to)e Fn(q)j Ft(disjoin)o(t)262 1154 y(dense)j(subsets)h Fn(S)552 1160 y Fl(1)571 1154 y Fn(;)7 b(:)g(:)g(:)t(;)g(S)688 1160 y Fj(q)707 1154 y Ft(,)18 b(and)g(that)g(an)o(y)f(t)o(w)o(o)h(suc) o(h)h(partitions)e(are)i(related)f(b)o(y)g(an)262 1204 y(order-preserving)j(p)q(erm)o(utation.)34 b(Let)20 b Fn(G)p Ft(\()p Fn(q)q Ft(\))g(b)q(e)g(the)h(group)e(of)g(p)q(erm)o (utations)g(of)g Fi(Q)262 1254 y Ft(whic)o(h)12 b(preserv)o(e)j(the)e (order)h(and)e(the)i(subsets)g Fn(S)1035 1260 y Fl(1)1054 1254 y Fn(;)7 b(:)g(:)g(:)e(;)i(S)1172 1260 y Fj(q)1190 1254 y Ft(.)18 b(An)13 b(orbit)f(of)g Fn(G)p Ft(\()p Fn(q)q Ft(\))h(on)g Fn(n)p Ft(-sets)262 1303 y(is)e(sp)q(eci\014ed)j(b) o(y)e(the)h(w)o(ord)f Fn(x)718 1309 y Fl(1)743 1303 y Fn(:)7 b(:)g(:)e(x)822 1309 y Fj(n)857 1303 y Ft(in)11 b(the)i(alphab)q(et)f Fn(A)g Ft(=)g Fm(f)p Ft(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(q)q Fm(g)p Ft(,)k(where)i Fn(x)1570 1309 y Fj(i)1596 1303 y Ft(is)f(the)262 1353 y(index)j(of)g(the)i(set)f (con)o(taining)f(the)h Fn(i)855 1338 y Fl(th)905 1353 y Ft(p)q(oin)o(t)f(of)g(the)i Fn(n)p Ft(-set)f(\(in)g(the)g(order)g (induced)h(b)o(y)262 1403 y Fi(Q)p Ft(\).)e(Ev)o(ery)f(w)o(ord)g(of)f (length)h Fn(n)g Ft(is)g(realised;)g(so)f Fn(f)1049 1409 y Fj(n)1072 1403 y Ft(\()p Fn(G)p Ft(\()p Fn(q)q Ft(\)\))f(=)g Fn(q)1265 1388 y Fj(n)1288 1403 y Ft(.)324 1453 y(No)o(w)g Fm(A)450 1438 y Fj(G)p Fl(\()p Fj(q)q Fl(\))533 1453 y Ft(is)g(the)h(algebra)f(spanned)h(b)o(y)f(the)h(set)g Fn(A)1167 1438 y Fk(\003)1199 1453 y Ft(of)f(all)f(w)o(ords)i(in)f(the) h(alphab)q(et)262 1503 y Fn(A)p Ft(;)19 b(m)o(ultiplication)c(of)i(t)o (w)o(o)g(w)o(ords)h(is)g(giv)o(en)g(b)o(y)f(the)i(sum)e(of)g(all)g(w)o (ords)h(obtained)g(b)o(y)262 1553 y(\\sh)o(u\017ing")d(them)i (together.)28 b(F)m(or)17 b(example,)f(using)g Fm(f)p Fn(a;)7 b(b)p Fm(g)16 b Ft(instead)h(of)g Fm(f)p Ft(1)p Fn(;)7 b Ft(2)p Fm(g)15 b Ft(for)i(the)262 1602 y(alphab)q(et,)c(w)o(e) h(ha)o(v)o(e)622 1694 y(\()p Fn(ab)p Ft(\))c Fm(\001)e Ft(\()p Fn(aab)p Ft(\))k(=)g Fn(abaab)d Ft(+)g(3)p Fn(aabab)f Ft(+)i(6)p Fn(aaabb:)262 1785 y Ft(This)15 b(is)h(the)h Fo(shu\017e)h(algebr)n(a)p Ft(,)d(whic)o(h)h(arises)h(in)f(the)g (theory)h(of)e(free)i(Lie)f(algebras)g(\(see)262 1835 y(Reutenauer)h([28)o(]\).)23 b(It)16 b(w)o(as)g(pro)o(v)o(ed)g(b)o(y)f (Radford)g([26)o(])g(that)h(the)h(sh)o(u\017e)f(algebra)f(on)h(a)262 1885 y(giv)o(en)10 b(alphab)q(et)h(is)g(a)g(p)q(olynomial)c(algebra)k (generated)h(b)o(y)f(the)h Fo(Lyndon)h(wor)n(ds)p Ft(.)k(In)11 b(order)262 1934 y(to)17 b(explain)f(these,)j(w)o(e)e(assume)g(that)g (the)h(alphab)q(et)f Fn(A)g Ft(is)g(totally)f(ordered,)j(and)e(tak)o(e) 262 1984 y(the)f(lexicographic)f(order)i(on)f(the)g(w)o(ords.)25 b(No)o(w)16 b(a)f Fo(Lyndon)j(wor)n(d)e Ft(is)f(a)h(w)o(ord)g(whic)o(h) g(is)262 2034 y(smaller)8 b(\(in)i(this)g(order\))h(than)f(an)o(y)g (prop)q(er)h(cyclic)f(shift)g(of)f(itself;)i(that)f(is,)g Fn(w)h Ft(is)f(a)g(Lyndon)262 2084 y(w)o(ord)k(if,)g(whenev)o(er)i Fn(w)e Ft(=)g Fn(xy)i Ft(is)f(a)g(prop)q(er)h(factorisation,)d(w)o(e)j (ha)o(v)o(e)e Fn(w)g(<)g(y)q(x)p Ft(.)21 b(No)o(w)15 b(the)262 2134 y(com)o(binatorial)10 b(assertions)15 b(required)g(for)f(Radford's)f(theorem)g(are)h(the)h(follo)o(wing:)282 2217 y(\(a\))h(an)o(y)g(w)o(ord)g(has)h(a)f(unique)g(expression)h(as)f (a)g(concatenation)h Fn(w)1378 2223 y Fl(1)1396 2217 y Fn(w)1426 2223 y Fl(2)1452 2217 y Fn(:)7 b(:)g(:)e(w)1537 2223 y Fj(n)1559 2217 y Ft(,)16 b(where)365 2267 y Fn(w)395 2273 y Fl(1)414 2267 y Fn(;)7 b(:)g(:)g(:)t(;)g(w)536 2273 y Fj(n)572 2267 y Ft(are)14 b(Lyndon)g(w)o(ords)g(and)g Fn(w)1023 2273 y Fl(1)1053 2267 y Fm(\025)d Fn(w)1126 2273 y Fl(2)1156 2267 y Fm(\025)h Fn(:)7 b(:)g(:)j Fm(\025)i Fn(w)1334 2273 y Fj(n)1356 2267 y Ft(;)282 2350 y(\(b\))e(of)f(all)f (the)h(w)o(ords)h(whic)o(h)f(can)h(b)q(e)f(obtained)h(b)o(y)f(sh)o (u\017ing)f(Lyndon)h(w)o(ords)h Fn(w)1533 2356 y Fl(1)1551 2350 y Fn(;)d(:)g(:)g(:)e(;)i(w)1674 2356 y Fj(n)365 2399 y Ft(together,)k(the)f(lexicographically)d(greatest)k(is)e(the)h (concatenation)f(in)g(non-increasing)365 2449 y(order.)957 2574 y(14)p eop %%Page: 15 15 15 14 bop 262 307 a Ft(No)o(w)14 b(w)o(e)h(tak)o(e)f(the)h (\\connected")h(w)o(ords)f(to)f(b)q(e)i(the)f(Lyndon)f(w)o(ords,)h(and) f(the)h(relation)262 357 y(of)c(\\in)o(v)o(olv)o(emen)o(t")f(to)j(b)q (e)g(lexicographic)f(order)h(rev)o(ersed;)i(and)d(this)h(result)g (\014ts)g(in)o(to)f(the)262 407 y(previous)i(formalism)o(.)324 457 y(Note)d(that)f(the)h(n)o(um)o(b)q(er)f(of)g(Lyndon)g(w)o(ords)g (of)g(length)h Fn(n)f Ft(is)1278 440 y Fl(1)p 1276 447 21 2 v 1276 471 a Fj(n)1309 425 y Fg(P)1352 469 y Fj(d)p Fk(j)p Fj(n)1409 457 y Fn(\026)p Ft(\()p Fn(d)p Ft(\))p Fn(q)1508 442 y Fj(n=d)1565 457 y Ft(,)g(where)262 506 y Fn(\026)16 b Ft(is)g(the)i(M\177)-21 b(obius)16 b(function.)25 b(This)17 b(is)f(a)h(w)o(ell-kno)o(wn)e(expression,)j(whic)o(h)e(also)g (coun)o(ts)262 556 y(\(among)c(other)i(things\))h(the)f(n)o(um)o(b)q (er)g(of)f(monic)g(irreducible)i(p)q(olynomial)o(s)d(of)h(degree)j Fn(n)262 606 y Ft(o)o(v)o(er)e(the)h(\014nite)g(\014eld)g(of)f(order)h Fn(q)q Ft(,)f(if)g Fn(q)h Ft(is)g(a)f(prime)g(p)q(o)o(w)o(er.)20 b(But)15 b(that)g(is)f(another)h(story)262 656 y(\(see)g(Bailey)e Fo(et)i(al.)e Ft([2]\).)262 793 y Fp(7)66 b(Reconstruction)262 884 y Ft(The)17 b(algebraic)g(considerations)h(of)e(the)i(last)f (section)h(are)f(also)g(related)h(to)f(the)h(v)o(ertex)262 934 y(reconstruction)c(conjecture)g(for)e(graphs.)18 b(View)o(ed)13 b(in)f(this)h(w)o(a)o(y)m(,)e(w)o(e)i(ha)o(v)o(e)f(a)h (reconstruc-)262 984 y(tion)h(problem)g(for)h(the)h(age)f(of)g(an)o(y)g (oligomo)o(rphic)e(group.)22 b(The)16 b(details)f(di\013er)h(greatly) 262 1034 y(from)c(one)i(class)g(to)g(another.)324 1083 y(Let)g Fn(G)e Ft(b)q(e)i(the)g(automorphism)c(group)k(of)e(the)i (random)e(graph,)h(so)g(that)g(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)262 1133 y(class)16 b(of)f Fn(G)h Ft(is)g(the)g(class)h(of)e(all)g(\014nite)h(graphs.)25 b(W)m(e)16 b(can)g(regard)g(the)h(v)o(ector)g(space)g Fn(V)1671 1139 y Fj(n)262 1183 y Ft(as)f(ha)o(ving)e(a)i(basis)g(whic)o (h)g(consists)h(of)e(the)h(isomorphism)d(t)o(yp)q(es)k(of)e Fn(n)p Ft(-v)o(ertex)i(graphs.)262 1233 y(Let)c Fn(T)359 1239 y Fj(n;n)p Fk(\000)p Fl(1)468 1233 y Ft(b)q(e)h(the)f(linear)g (map)e(from)h Fn(V)922 1239 y Fj(n)958 1233 y Ft(to)h Fn(V)1032 1239 y Fj(n)p Fk(\000)p Fl(1)1110 1233 y Ft(whic)o(h)g(tak)o (es)g(eac)o(h)h Fn(n)p Ft(-v)o(ertex)g(graph)262 1283 y(to)i(the)i(sum)e(of)g(its)h(\()p Fn(n)12 b Fm(\000)f Ft(1\)-v)o(ertex)18 b(induced)g(subgraphs.)28 b(Then)17 b Fn(T)1383 1289 y Fj(n;n)p Fk(\000)p Fl(1)1496 1283 y Ft(is)g(the)g(map)262 1332 y(represen)o(ted)d(b)o(y)e(the)h(matrix)d Fn(M)17 b Ft(of)11 b(Section)i(2;)f(its)g(dual)f(is)h(the)h(map)d Fn(T)1400 1338 y Fj(n)p Fk(\000)p Fl(1)p Fj(;n)1508 1332 y Ft(from)g Fn(V)1628 1338 y Fj(n)p Fk(\000)p Fl(1)262 1382 y Ft(to)i Fn(V)335 1388 y Fj(n)371 1382 y Ft(induced)i(b)o(y)f(m)o (ultiplicatio)o(n)e(b)o(y)h(the)i(elemen)o(t)f Fn(e)g Ft(of)g(the)g(preceding)h(section,)g(with)262 1432 y(matrix)e Fn(P)19 b Ft(as)14 b(in)f(Section)h(2.)324 1482 y(No)o(w)k(t)o(w)o(o)g Fn(n)p Ft(-v)o(ertex)h(graphs)g(are)g Fo(hyp)n(omorphic)g Ft(if)f(they)h(ha)o(v)o(e)f(the)h(same)f(dec)o(k)h(of)262 1532 y(v)o(ertex-deleted)d(subgraphs;)f(that)g(is,)f(if)g(their)h (images)e(under)i Fn(T)1303 1538 y Fj(n;n)p Fk(\000)p Fl(1)1413 1532 y Ft(are)g(equal.)20 b(So)14 b(if)262 1581 y Fn(X)20 b Ft(and)c Fn(Y)26 b Ft(are)16 b(h)o(yp)q(omorphic,)f (then)i Fn(X)e Fm(\000)c Fn(Y)25 b Fm(2)16 b Ft(k)o(er\()p Fn(T)1162 1587 y Fj(n;n)p Fk(\000)p Fl(1)1258 1581 y Ft(\).)26 b(Moreo)o(v)o(er,)17 b(for)f(an)o(y)g Fn(X)262 1631 y Ft(and)e Fn(Y)9 b Ft(,)14 b(if)f Fn(aX)h Ft(+)9 b Fn(bY)22 b Fm(2)12 b Ft(k)o(er\()p Fn(T)750 1637 y Fj(n;n)p Fk(\000)p Fl(1)846 1631 y Ft(\),)i(with)g Fn(ab)e Fm(6)p Ft(=)h(0,)g(then)i Fn(b)e Ft(=)f Fm(\000)p Fn(a)p Ft(,)i(and)g Fn(X)k Ft(and)d Fn(Y)23 b Ft(are)262 1681 y(h)o(yp)q(omorphic.)324 1731 y(So)12 b(the)h Fo(vertex)g(r)n(e)n(c)n (onstruction)h(c)n(onje)n(ctur)n(e)e Ft(for)g(graphs)h(can)f(b)q(e)h (stated)h(in)e(the)h(form:)262 1781 y Fo(F)m(or)19 b Fn(n)h(>)h Ft(2)p Fo(,)f(the)g(kernel)g(of)g Fn(T)777 1787 y Fj(n;n)p Fk(\000)p Fl(1)892 1781 y Fo(has)g(minimum)g(weight)f (gr)n(e)n(ater)g(than)h Ft(2.)34 b(\(The)262 1831 y Fo(minimum)16 b(weight)f Ft(of)g(a)g(subspace,)i(as)f(in)f(co)q(ding)g(theory)m(,)h (is)f(the)h(smallest)e(n)o(um)o(b)q(er)h(of)262 1880 y(non-zero)f(co)q(ordinates)h(of)e(a)g(non-zero)i(v)o(ector)g(in)e (that)h(subspace.\))324 1930 y(W)m(e)9 b(could)g(th)o(us)h(ask)f(the)h (question:)17 b Fo(What)11 b(is)f(the)h(minimum)g(weight)f(of)h Ft(k)o(er\()p Fn(T)1560 1936 y Fj(n;n)p Fk(\000)p Fl(1)1656 1930 y Ft(\))p Fo(?)262 1980 y Ft(F)m(or)k(example,)f(a)i(trivial)e (upp)q(er)j(b)q(ound)e(for)h(the)g(minim)n(um)c(w)o(eigh)o(t)j(is)g(1)c (+)f Fn(n=)p Ft(2)15 b(if)g Fn(n)h Ft(is)262 2030 y(ev)o(en.)i(F)m(or,) 13 b(if)g Fn(X)530 2036 y Fj(n;k)595 2030 y Ft(is)h(the)h(graph)e(with) h Fn(n)g Ft(v)o(ertices)h(and)f Fn(k)g Ft(disjoin)o(t)f(edges,)i(then) 369 2121 y Fm(h)p Fn(X)419 2127 y Fj(n;)p Fl(0)468 2121 y Fn(;)7 b(X)521 2127 y Fj(n;)p Fl(1)570 2121 y Fn(;)g(:)g(:)g(:)e(;)i (X)697 2128 y Fj(n;n=)p Fl(2)783 2121 y Fm(i)p Fn(T)823 2127 y Fj(n;n)p Fk(\000)p Fl(1)931 2121 y Fm(\022)k(h)p Fn(X)1024 2127 y Fj(n)p Fk(\000)p Fl(1)p Fj(;)p Fl(0)1116 2121 y Fn(;)c(X)1169 2127 y Fj(n)p Fk(\000)p Fl(1)p Fj(;)p Fl(1)1261 2121 y Fn(;)g(:)g(:)g(:)e(;)i(X)1388 2128 y Fj(n)p Fk(\000)p Fl(1)p Fj(;n=)p Fl(2)p Fk(\000)p Fl(1)1559 2121 y Fm(i)p Fn(:)262 2212 y Ft(So)12 b(some)g(non-zero)h(elemen)o(t)f (in)g Fm(h)p Fn(X)837 2218 y Fj(n;)p Fl(0)886 2212 y Fn(;)7 b(:)g(:)g(:)e(;)i(X)1013 2219 y Fj(n;n=)p Fl(2)1100 2212 y Fm(i)12 b Ft(b)q(elongs)h(to)f(the)h(k)o(ernel)g(of)f Fn(T)1586 2218 y Fj(n;n)p Fk(\000)p Fl(1)1682 2212 y Ft(.)262 2262 y(This)d(can)h(surely)h(b)q(e)f(impro)o(v)o(ed;)f(but)h (is)g(the)h(minim)n(um)5 b(w)o(eigh)o(t)10 b(b)q(ounded)g(b)o(y)g(an)g (absolute)262 2312 y(constan)o(t?)324 2362 y(W)m(e)f(can)G(广义)h(and)e(问:)15 b Fo(什么)d(is)e(h)(最小)f(权重)g(of)h FT(k)o(ER)(p fn(t)1618 2368 y fj(n;m)1681 2362 y ft(\))262≤2412 yFo(for)j fn(m)f(<)g(n)pFo(?)20 b Ft(\(W)m(e)14 b(de\014ne)i Fn(T)721 2418 y Fj(n;m)797 2412 y Ft(to)f(b)q(e)g(the)g (linear)f(map)f(taking)g(an)h Fn(n)p Ft(-v)o(ertex)h(graph)g(to)957 2574 y(15)p eop %%Page: 16 16 16 15 bop 262 307 a Ft(the)14 b(sum)f(of)g(its)h Fn(m)p Ft(-v)o(ertex)h(subgraphs.\))k(Since)748 420 y Fn(T)772 426 y Fj(n;l)816 420 y Fn(T)840 426 y Fj(l;m)904 420 y Ft(=)947 361 y Fg(\022)978 392 y Fn(n)9 b Fm(\000)h Fn(m)984 448 y(l)g Fm(\000)g Fn(m)1090 361 y Fg(\023)1121 420 y Fn(T)1145 426 y Fj(n;m)262 533 y Ft(for)k Fn(m)g(<)g(l)h(<)e(n)p Ft(,)i(the)h(minim)n(um)10 b(w)o(eigh)o(t)15 b(of)g(k)o(er\()p Fn(T)1082 539 y Fj(n;m)1144 533 y Ft(\))g(decreases)j(as)d Fn(m)h Ft(decreases.)24 b Fo(Is)262 583 y(ther)n(e)16 b(an)i(absolute)f(c)n(onstant)h Fn(k)g Fo(such)f(that)g Ft(k)o(er)q(\()p Fn(T)1077 589 y Fj(n;n)p Fk(\000)p Fj(k)1174 583 y Ft(\))h Fo(has)f(minimum)g(weight)f Ft(2)h Fo(for)262 632 y(al)r(l)d Fn(n)p Fo(?)f(HF)q(EN)G(IF)F(W)O(E)262 756 Y(W)O(ORK)E((1)I(O)O(V)O(V)O(ER)F(A)G(\014ED)G(OF)F(非零)I(C)O(HaCr)(G)FN(P)F FT(\(SUC)O(H)H(AS)F(H)O(TEN)G(Mo)Q(D)262,805 Y Y FN(P)P FT(\)?)324 706 Y FT(TW)O(O)H(进一步)I(广义)F(建议)G(MeSELV)O(ES)24 B(第一,)j(If)13 b Fn(p)h Ft(divides)f Fn(n)p Ft(,)g(then)i(k)o(er\()p Fn(T)792 811 y Fj(n;n)p Fk(\000)p Fl(1)888 805 y Ft(\))f(has)g(minim)n(um)9 b(w)o(eigh)o(t)14 b(1:)j(an)o(y)d(graph)f(with)h(all)262 855 y(its)e(v)o(ertex-deleted)i (subgraphs)g(isomorphic)d(b)q(elongs)h(to)h(the)g(k)o(ernel)g(\(for)f (example,)f(an)o(y)262 905 y(v)o(ertex-transitiv)o(e)j(graph\).)324 955 y(Second,)20 b(these)g(questions)g(can)f(b)q(e)g(p)q(osed)h(for)e (other)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)20 b(\(or)e(more)g(general\))262 1005 y(classes)d(of)e(structures.)20 b(As)15 b(an)e(example,)g(consider)h(strings)h(of)e(length)h Fn(n)g Ft(o)o(v)o(er)f(a)h(binary)262 1054 y(alphab)q(et)h Fm(f)p Fn(a;)7 b(b)p Fm(g)p Ft(.)21 b(As)16 b(earlier,)g(w)o(e)f (consider)i(these)g(as)e(sets)i(with)e(a)g(total)g(order)h(whose)262 1104 y(elemen)o(ts)e(are)h(partitioned)f(in)o(to)g(t)o(w)o(o)g (distinguished)h(subsets.)22 b(So)14 b(a)g(substructure)j(is)e(a)262 1154 y(\(not)f(necessarily)i(consecutiv)o(e\))g(substring.)k(The)15 b(class)g(of)f(suc)o(h)i(strings)f(is)f(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)262 1204 y(class)18 b(of)g(the)h(group)f Fn(G)p Ft(\(2\))g(of)g(order-preserving)i(p)q(erm)o(utations)d(of)h Fi(Q)e Ft(whic)o(h)i(\014x)g(t)o(w)o(o)262 1254 y(complemen)o(tary)11 b(dense)k(subsets.)324 1303 y(No)o(w)i Fn(T)446 1309 y Fj(n;m)526 1303 y Ft(maps)f(a)h(string)h(to)f(the)h(sum)f(of)g(its)h Fn(m)p Ft(-elemen)o(t)f(substrings,)i(coun)o(ted)262 1353 y(with)12 b(m)o(ultipliciti)o(es.)k(Call)c(t)o(w)o(o)g(strings)h Fn(u)g Ft(and)f Fn(v)j(m)p Fo(-e)n(quivalent)e Ft(if)f(they)h(ha)o(v)o (e)g(the)g(same)262 1403 y(image;)f(that)j(is,)f(if)g(eac)o(h)h(string) f(of)g(length)h Fn(m)g Ft(has)g(the)g(same)e(m)o(ultiplicit)o(y)f(in)i Fn(u)g Ft(and)h Fn(v)q Ft(.)262 1453 y(\(This)d(can)h(b)q(e)h(extended) g(to)e(strings)i(of)e(length)g(less)i(than)e Fn(m)i Ft(b)o(y)e (de\014ning)h(suc)o(h)g(a)g(string)262 1503 y(to)j(b)q(e)g Fn(m)p Ft(-equiv)n(alen)o(t)g(only)f(to)h(itself.\))25 b(F)m(or)16 b(example,)f(the)h(strings)h Fn(X)i Ft(=)d Fn(abbbaab)f Ft(and)262 1553 y Fn(Y)20 b Ft(=)12 b Fn(baabbba)h Ft(of)g(length)h(7)g(are)g(3-equiv)n(alen)o(t,)e(since)j Fn(T)1148 1559 y Fl(7)p Fj(;)p Fl(3)1207 1553 y Ft(maps)e(b)q(oth)h Fn(X)j Ft(and)d Fn(Y)23 b Ft(to)483 1640 y Fn(aaa)9 b Ft(+)h(3)p Fn(aab)e Ft(+)i(6)p Fn(aba)e Ft(+)i(6)p Fn(abb)f Ft(+)g(3)p Fn(baa)g Ft(+)g(6)p Fn(bab)g Ft(+)g(6)p Fn(bba)g Ft(+)g(4)p Fn(bbb:)324 1728 y Ft(No)o(w)i(the)i(ob)o(vious)e(question)i (is:)k Fo(What)c(is)g(the)h(smal)r(lest)e Fn(n)p Fo(,)h(as)g(a)h (function)f(of)h Fn(m)p Fo(,)f(for)262 1778 y(which)h(ther)n(e)g(ar)n (e)g(two)g Fn(m)p Fo(-e)n(quivalent)h(binary)g(strings)f(of)g(length)h Fn(n)p Fo(?)j Ft(The)c(answ)o(er)g(is)g(not)262 1828 y(kno)o(wn,)e(and)h(the)h(kno)o(wn)e(upp)q(er)i(and)f(lo)o(w)o(er)g(b)q (ounds)h(are)f(v)o(ery)h(far)f(apart.)k(John)d(Dixon)262 1878 y([11)o(])f(pro)o(v)o(ed)h(a)g(result)h(c)o(haracterising)f Fn(m)p Ft(-equiv)n(alence)h(in)e(purely)h(algebraic)g(terms.)k(He)262 1927 y(sho)o(w)o(ed)12 b(that)g(t)o(w)o(o)f(strings)h(are)h Fn(m)p Ft(-equiv)n(alen)o(t)e(if)g(and)h(only)f(if,)g(when)i(regarded)f (as)g(w)o(ords)262 1977 y(in)h(the)h(generators)i(of)d(the)h(free)h (nilp)q(oten)o(t)f(group)f(of)h(class)g Fn(m)p Ft(,)g(they)g(are)g (equal.)324 2051 y(The)i Fo(e)n(dge)i(r)n(e)n(c)n(onstruction)e(c)n (onje)n(ctur)n(e)g Ft(for)g(graphs)h(can)f(b)q(e)h(\014tted)g(in)o(to)e (this)i(form-)262 2100 y(alism)12 b(to)j(some)f(exten)o(t)h(as)g(w)o (ell.)20 b(Let)15 b Fn(G)g Ft(b)q(e)g(the)g(symmetric)e(group)i(on)g (an)f(in\014nite)h(set)262 2150 y(\(sa)o(y)i Fi(N)p Ft(\),)g(in)g(its)h (induced)g(action)f(on)h(the)g(set)h(\012)f(=)1134 2117 y Fg(\000)1153 2132 y Fb(N)1155 2165 y Fl(2)1174 2117 y Fg(\001)1211 2150 y Ft(of)f(2-elemen)o(t)g(subsets)i(of)e Fi(N)p Ft(.)262 2200 y(No)o(w)d(an)g Fn(n)p Ft(-elemen)o(t)g(mem)o(b)q (er)f(of)h(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)f(of)g Fn(G)h Ft(consists)g(of)f(a)g(graph)h(with)f Fn(n)262 2250 y Ft(edges)j(\(in)f(other)h(w)o(ords,)g(an)g Fn(n)p Ft(-v)o(ertex)g(graph)f(whic)o(h)h(is)f(a)g(line)g(graph,)h(in)f (a)g(sp)q(eci\014ed)262 2300 y(w)o(a)o(y:)h(so)d(the)g(triangle)f(coun) o(ts)i(t)o(wice,)e(according)h(as)g(it)g(is)f(the)i(line)e(graph)g(of)h (a)f(triangle)262 2350 y(or)j(of)f(a)h(star\).)25 b(The)16 b(edge-reconstruction)i(conjecture)g(asserts)g(that)e(k)o(er\()p Fn(T)1505 2356 y Fj(n;n)p Fk(\000)p Fl(1)1601 2350 y Ft(\))g(has)262 2399 y(minim)n(um)11 b(w)o(eigh)o(t)16 b(greater)h(than)f(2)f(in)h(this)g(class,)g(pro)o(vided)g(that)g Fn(n)f(>)h Ft(3.)24 b(Questions)262 2449 y(lik)o(e)13 b(those)h(p)q(osed)h(earlier)f(for)g(v)o(ertex-reconstruction)i(can)e (no)o(w)g(b)q(e)g(ask)o(ed.)957 2574 y(16)p eop %%Page: 17 17 17 16 bop 324 307 a Ft(There)15 b(are)g(further)h(links)d(b)q(et)o(w)o (een)j(edge-reconstruction)h(and)d(\014nite)h(p)q(erm)o(utation)262 357 y(groups;)e(but)h(that)g(is)g(another)g(story)m(.)262 494 y Fp(8)66 b(Cycle)23 b(index)262 585 y Ft(No)o(w)16 b(w)o(e)g(come)g(to)g(the)h(rule)g(for)f(calculating)f(the)i(sequence)i (op)q(erator)e(corresp)q(onding)262 635 y(to)g(an)o(y)g(oligomorphic)e (group.)30 b(W)m(e)17 b(will)g(also)g(see)i(ho)o(w)e(to)h(coun)o(t)g (orbits)g(on)f(ordered)262 685 y Fn(n)p Ft(-tuples)c(of)f(distinct)i (elemen)o(ts)f(\(whic)o(h)g(amoun)o(ts)e(to)i(the)h(same)e(thing)h(as)g (en)o(umerating)262 735 y(lab)q(elled)g(structures)j(in)e(the)g(F)m(ra) -5 b(\177)-16 b(\020ss)o(\023)c(e)14 b(class)h(of)e(the)i(group\).)324 784 y(W)m(e)e(b)q(egin)h(with)g(a)f(little)g(P\023)-21 b(oly)o(a)13 b(theory)m(.)18 b(Let)c(\012)g(b)q(e)h(a)e(\014nite)h(set) h(of)e(size)i Fn(n)p Ft(.)j(F)m(or)13 b(an)o(y)262 840 y(p)q(erm)o(utation)8 b Fn(g)k Ft(of)d(\012,)i(w)o(e)f(de\014ne)h(the)g Fo(cycle)g(index)g Fn(z)r Ft(\()p Fn(g)q Ft(\))g(of)e Fn(g)j Ft(to)e(b)q(e)g Fn(s)1347 818 y Fj(c)1362 822 y Fc(1)1379 818 y Fl(\()p Fj(g)q Fl(\))1347 851 y(1)1424 840 y Fn(s)1443 818 y Fj(c)1458 822 y Fc(2)1475 818 y Fl(\()p Fj(g)q Fl(\))1443 851 y(2)1527 840 y Fm(\001)d(\001)g(\001)e Fn(s)1601 818 y Fj(c)1616 822 y Ff(n)1637 818 y Fl(\()p Fj(g)q Fl(\))1601 845 y Fj(n)1682 840 y Ft(,)262 890 y(where)14 b Fn(s)400 896 y Fl(1)419 890 y Fn(;)7 b(s)457 896 y Fl(2)476 890 y Fn(;)g(:)g(:)g(:)t(;)g(s)587 896 y Fj(n)623 890 y Ft(are)14 b(indep)q(enden)o(t)g(indeterminates,)f(and) g Fn(c)1317 896 y Fj(i)1331 890 y Ft(\()p Fn(g)q Ft(\))h(is)f(the)h(n)o (um)o(b)q(er)e(of)262 940 y(cycles)j(of)f(length)g Fn(i)h Ft(in)f(the)h(cycle)h(decomp)q(osition)d(of)h Fn(g)q Ft(.)20 b(If)14 b Fn(G)g Ft(is)g(a)g(p)q(erm)o(utation)g(group)262 989 y(on)f(\012,)h(the)g Fo(cycle)h(index)f Ft(of)g Fn(G)f Ft(is)h(the)g(a)o(v)o(erage)g(of)g(the)g(cycle)h(indices)f(of)f(its)h (elemen)o(ts:)786 1098 y Fn(Z)s Ft(\()p Fn(G)p Ft(\))e(=)960 1070 y(1)p 943 1088 56 2 v 943 1127 a Fm(j)p Fn(G)p Fm(j)1013 1059 y Fg(X)1010 1148 y Fj(g)q Fk(2)p Fj(G)1083 1098 y Fn(z)r Ft(\()p Fn(g)q Ft(\))p Fn(:)262 1232 y Ft(The)i(role)g(of)f (the)h(cycle)h(index)f(in)f(en)o(umeration)g(problems)g(is)h(w)o (ell-kno)o(wn.)324 1282 y(Clearly)e(it)g(is)h(imp)q(ossible)e(to)i (de\014ne)g(the)h(cycle)f(index)g(of)f(an)h(in\014nite)f(group)h(b)o(y) f(an)o(y-)262 1332 y(thing)d(lik)o(e)h(this)g(form)o(ula;)f(so)h(w)o(e) h(adopt)f(a)g(di\013eren)o(t)h(approac)o(h.)17 b(Let)11 b Fn(G)f Ft(b)q(e)h(oligomorphic.)262 1381 y(Cho)q(ose)j(represen)o (tativ)o(es)h(for)f(the)g(orbits)f(of)g Fn(G)g Ft(on)h(\014nite)g (subsets)h(of)e(\012.)18 b(F)m(or)13 b(eac)o(h)h(suc)o(h)262 1431 y(represen)o(tativ)o(e)i(\001,)d(let)h Fn(H)s Ft(\(\001\))g(b)q(e) h(the)f(group)g(induced)h(on)f(\001)f(b)o(y)h(its)g(set)o(wise)i (stabiliser)262 1481 y(in)d Fn(G)p Ft(.)18 b(No)o(w)13 b(de\014ne)i(the)g Fo(mo)n(di\014e)n(d)g(cycle)g(index)1043 1470 y Ft(~)1034 1481 y Fn(Z)t Ft(\()p Fn(G)p Ft(\))e(of)h Fn(G)f Ft(to)h(b)q(e)787 1570 y(~)778 1581 y Fn(Z)t Ft(\()p Fn(G)p Ft(\))d(=)930 1541 y Fg(X)946 1630 y Fl(\001)997 1581 y Fn(Z)s Ft(\()p Fn(H)s Ft(\(\001\)\))p Fn(;)262 1709 y Ft(where)18 b(the)f(sum)f(is)h(o)o(v)o(er)g(the)g(orbit)g (represen)o(tativ)o(es.)29 b(This)17 b(is)g(meaningful,)d(since)k(b)o (y)262 1759 y(assumption)11 b(there)k(are)e(only)g(\014nitely)g(man)o (y)e(orbits)i(of)g(size)h Fn(n)p Ft(,)f(and)g(hence)h(a)f(monomi)o(al) 262 1808 y(of)j(w)o(eigh)o(t)i Fn(n)f Ft(o)q(ccurs)i(only)e(\014nitely) g(man)o(y)f(times)g(in)h(the)i(sum)d(\(where)j(the)f(w)o(eigh)o(t)f(of) 262 1858 y Fn(s)281 1840 y Fj(c)296 1844 y Fc(1)281 1869 y Fl(1)314 1858 y Fn(s)333 1840 y Fj(c)348 1844 y Fc(2)333 1869 y Fl(2)374 1858 y Fm(\001)7 b(\001)g(\001)e Fn(s)448 1843 y Fj(c)463 1847 y Ff(n)448 1869 y Fj(n)499 1858 y Ft(is)14 b(de\014ned)h(to)f(b)q(e)g Fn(c)809 1864 y Fl(1)837 1858 y Ft(+)c(2)p Fn(c)918 1864 y Fl(2)945 1858 y Ft(+)g Fm(\001)d(\001)g(\001)h Ft(+)h Fn(nc)1129 1864 y Fj(n)1151 1858 y Ft(\).)324 1908 y(This)14 b(pro)q(cedure)j(is)e (meaningful)d(for)i(\014nite)h(groups)g Fn(G)p Ft(,)g(but)g(it)f(giv)o (es)h(nothing)f(new:)262 1958 y(in)h(fact,)i(for)e(a)h(\014nite)h (group)f Fn(G)p Ft(,)809 1947 y(~)801 1958 y Fn(Z)s Ft(\()p Fn(G)p Ft(\))g(is)g(obtained)g(from)e Fn(Z)s Ft(\()p Fn(G)p Ft(\))j(b)o(y)f(the)g(substitution)262 2008 y(replacing)c Fn(s)457 2014 y Fj(i)484 2008 y Ft(b)o(y)g Fn(s)559 2014 y Fj(i)581 2008 y Ft(+)7 b(1)12 b(for)h(all)e Fn(i)p Ft(.)18 b(\(F)m(or)12 b(exp)q(erts)j(in)d(P\023)-21 b(oly)o(a)12 b(theory)m(,)g(this)h(is)g(an)f(exercise.\))324 2058 y(I)g(no)o(w)g(list)f(three)j(pairs)e(of)f(facts)i(ab)q(out)f(the)g(mo) q(di\014ed)f(cycle)i(index:)k(\014rst,)c(its)f(v)n(alues)262 2107 y(for)i(the)i(groups)g Fn(S)i Ft(and)d Fn(A)p Ft(;)h(second,)g (its)f(b)q(eha)o(viour)g(under)h(taking)f(direct)h(and)f(wreath)262 2157 y(pro)q(ducts;)g(and)g(third,)f(a)g(couple)h(of)f(in)o(teresting)h (sp)q(ecialisations)f(of)g(it.)20 b(First,)14 b(another)262 2207 y(de\014nition.)20 b(If)15 b Fn(G)f Ft(is)h(oligomo)o(rphic)e(on)h (\012,)h(w)o(e)g(let)g Fn(F)1111 2213 y Fj(n)1133 2207 y Ft(\()p Fn(G)p Ft(\))g(b)q(e)g(the)h(n)o(um)o(b)q(er)e(of)g Fn(G)p Ft(-orbits)262 2257 y(on)g Fn(n)p Ft(-tuples)g(of)g(distinct)g (elemen)o(ts)g(of)g(\012.)19 b(The)c(\014niteness)g(of)f(this)g(n)o(um) o(b)q(er)g(for)g(all)f Fn(n)h Ft(is)262 2307 y(equiv)n(alen)o(t)f(to)g (the)i(oligomorph)o(y)c(of)i Fn(G)p Ft(;)g(indeed,)h(w)o(e)g(ha)o(v)o (e)836 2398 y Fn(f)856 2404 y Fj(n)891 2398 y Fm(\024)e Fn(F)962 2404 y Fj(n)996 2398 y Fm(\024)f Fn(n)p Ft(!)p Fn(f)1096 2404 y Fj(n)957 2574 y Ft(17)p eop %%Page: 18 18 18 17 bop 262 307 a Ft(for)11 b(all)g Fn(n)p Ft(.)17 b(If)12 b Fn(G)g Ft(is)g(the)g(automorphism)d(group)j(of)g(a)g (homogeneous)f(relational)f(structure)262 357 y Fn(X)s Ft(,)h(then)g Fn(F)440 363 y Fj(n)462 357 y Ft(\()p Fn(G)p Ft(\))f(is)g(the)h(n)o(um)o(b)q(er)f(of)g(lab)q(elled)f Fn(n)p Ft(-elemen)o(t)h(structures)j(in)d(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)10 b(class)262 407 y(\(that)i(is,)g(the) g(n)o(um)o(b)q(er)g(of)f(structures)k(on)d(the)h(set)g Fm(f)p Ft(1)p Fn(;)7 b Ft(2)p Fn(;)g(:)g(:)g(:)s(;)g(n)p Fm(g)12 b Ft(whic)o(h)g(are)g(em)o(b)q(eddable)262 457 y(in)j Fn(X)s Ft(\).)27 b(As)16 b(standard)h(in)f(en)o(umeration)f (theory)m(,)i(w)o(e)f(describ)q(e)j(the)d(sequence)j(\()p Fn(F)1595 463 y Fj(n)1617 457 y Ft(\))e(b)o(y)262 506 y(an)c Fo(exp)n(onential)j(gener)n(ating)f(function)f Ft(giv)o(en)g(b)o(y)778 629 y Fn(F)805 635 y Fj(G)832 629 y Ft(\()p Fn(t)p Ft(\))e(=)950 577 y Fk(1)936 589 y Fg(X)935 677 y Fj(n)p Fl(=0)1009 601Y-Fn(f)1036(607)y FJ(n)1059 601 y FT(\)(p fn(g)p ft(\))p fn(t)1139 586πy fJ(n)p 1009 619 152 152 2 1067 1067 a fn(n)p ft(!)1166 629 y Fn(:)324 812 y Fm(\017)374 802 y Ft(~)365 812 y Fn(Z)t Ft(\()p Fn(S)r Ft(\))g(=)g(exp)582 729 y Fg(0)582 803 y(@)632 760 y Fk(1)619 773 y Fg(X)620 861 y Fj(j)r Fl(=1)691 784 y Fn(s)710 790 y Fj(j)p 691 803 37 2 v 699 841 a Fn(j)733 729 y Fg(1)733 803 y(A)769 812 y Fn(:)324 969 y Fm(\017)374 958 y Ft(~)365 969 y Fn(Z)t Ft(\()p Fn(A)p Ft(\))g(=)565 941 y(1)p 520 959 110 2 v 520 997 a(1)d Fm(\000)h Fn(s)611 1003 y Fl(1)635 969 y Fn(:)324 1079 y Fm(\017)374 1068 y Ft(~)365 1079 y Fn(Z)t Ft(\()p Fn(H)i Fm(\002)d Fn(K)s Ft(\))j(=)620 1068 y(~)611 1079 y Fn(Z)s Ft(\()p Fn(H)s Ft(\))721 1068 y(~)712 1079 y Fn(Z)t Ft(\()p Fn(K)s Ft(\))p Fn(:)324 1162 y Fm(\017)374 1151 y Ft(~)365 1162 y Fn(Z)t Ft(\()p Fn(H)e Ft(W)m(r)c Fn(K)s Ft(\))16 b(is)g(obtained)g(from)916 1151 y(~)907 1162 y Fn(Z)s Ft(\()p Fn(K)s Ft(\))g(b)o(y)g(substituting)1326 1151 y(~)1317 1162 y Fn(Z)t Ft(\()p Fn(H)s Ft(\)\()p Fn(s)1454 1168 y Fj(i)1468 1162 y Fn(;)7 b(s)1506 1168 y Fl(2)p Fj(i)1536 1162 y Fn(;)g(:)g(:)g(:)n Ft(\))k Fm(\000)g Ft(1)365 1212 y(for)j Fn(s)448 1218 y Fj(i)462 1212 y Ft(,)f(for)h Fn(i)e Ft(=)g(1)p Fn(;)7 b Ft(2)p Fn(;)g(:)g(:)g(:)s(:)324 1295 y Fm(\017)20 b Fn(f)385 1301 y Fj(G)414 1295 y Ft(\()p Fn(t)p Ft(\))14 b(is)f(obtained)h(from) 794 1284 y(~)785 1295 y Fn(Z)s Ft(\()p Fn(G)p Ft(\))g(b)o(y)g (substituting)g Fn(t)1199 1280 y Fj(i)1227 1295 y Ft(for)f Fn(s)1309 1301 y Fj(i)1337 1295 y Ft(for)h Fn(i)e Ft(=)f(1)p Fn(;)c Ft(2)p Fn(;)g(:)g(:)g(:)m Ft(.)324 1378 y Fm(\017)20 b Fn(F)392 1384 y Fj(G)420 1378 y Ft(\()p Fn(t)p Ft(\))g(is)g(obtained) f(from)824 1367 y(~)815 1378 y Fn(Z)t Ft(\()p Fn(G)p Ft(\))g(b)o(y)h(substituting)g Fn(t)g Ft(for)f Fn(s)1355 1384 y Fl(1)1394 1378 y Ft(and)h(0)f(for)h Fn(s)1610 1384 y Fj(i)1644 1378 y Ft(for)365 1428 y Fn(i)12 b Ft(=)g(2)p Fn(;)7 b Ft(3)p Fn(;)g(:)g(:)g(:)m Ft(.)324 1511 y(It)16 b(follo)o(ws)e(from)f(the)k(direct)f(pro)q(duct)h(rule)f(and)f(the)h(t) o(w)o(o)f(sp)q(ecialisations)h(that,)g(as)262 1560 y(w)o(ell)i(as)h Fn(f)427 1566 y Fj(H)r Fk(\002)p Fj(K)515 1560 y Ft(\()p Fn(t)p Ft(\))i(=)f Fn(f)655 1566 y Fj(H)687 1560 y Ft(\()p Fn(t)p Ft(\))p Fn(f)754 1566 y Fj(K)787 1560 y Ft(\()p Fn(t)p Ft(\),)g(w)o(e)f(also)g(ha)o(v)o(e)g Fn(F)1149 1566 y Fj(H)r Fk(\002)p Fj(K)1236 1560 y Ft(\()p Fn(t)p Ft(\))i(=)f Fn(F)1383 1566 y Fj(H)1414 1560 y Ft(\()p Fn(t)p Ft(\))p Fn(F)1488 1566 y Fj(K)1520 1560 y Ft(\()p Fn(t)p Ft(\).)35 b(But,)262 1610 y(b)q(ecause)22 b(these)f(are)g(exp)q (onen)o(tial)f(generating)h(functions,)g(the)g(con)o(v)o(olution)e (rule)i(for)262 1660 y(sequences)16 b(is)e(a)f(little)g(di\013eren)o (t,)i(namely)623 1782 y Fn(F)650 1788 y Fj(n)672 1782 y Ft(\()p Fn(H)d Fm(\002)e Fn(K)s Ft(\))i(=)906 1730 y Fj(n)887 1743 y Fg(X)886 1832 y Fj(k)q Fl(=0)954 1724 y Fg(\022)984 1754 y Fn(n)985 1811 y(k)1009 1724 y Fg(\023)1040 1782 y Fn(F)1067 1788 y Fj(k)1087 1782 y Ft(\()p Fn(H)s Ft(\))p Fn(F)1184 1788 y Fj(n)p Fk(\000)p Fj(k)1251 1782 y Ft(\()p Fn(K)s Ft(\))p Fn(:)262 1911 y Ft(This)h(is)h(the)h (so-called)e Fo(exp)n(onential)j(c)n(onvolution)p Ft(.)324 1961 y(The)k(\014fth)g(of)f(the)h(six)g(p)q(oin)o(ts)f(giv)o(es)h(us)g (the)g(rule)g(for)f(calculating)g(the)h(sequence)262 2011 y(\()p Fn(f)298 2017 y Fj(n)321 2011 y Ft(\()p Fn(H)10 b Ft(W)m(r)c Fn(K)s Ft(\)\))13 b(from)f(\()p Fn(f)661 2017 y Fj(n)684 2011 y Ft(\()p Fn(H)s Ft(\)\):)18 b Fn(f)820 2017 y Fj(H)8 b Fl(W)m(r)e Fj(K)936 2011 y Ft(\()p Fn(t)p Ft(\))14 b(is)f(obtained)f(from)1313 2000 y(~)1305 2011 y Fn(Z)s Ft(\()p Fn(K)s Ft(\))i(b)o(y)e(substituting)262 2060 y Fn(f)282 2066 y Fj(H)313 2060 y Ft(\()p Fn(t)344 2045 y Fj(i)358 2060 y Ft(\))g Fm(\000)h Ft(1)k(for)h Fn(s)556 2066 y Fj(i)570 2060 y Ft(,)h(for)f Fn(i)g Ft(=)h(1)p Fn(;)7 b Ft(2)p Fn(;)g(:)g(:)g(:)m Ft(.)30 b(W)m(e)18 b(see)h(that)f(the)h(information)c(ab)q(out)j Fn(K)j Ft(w)o(e)262 2110 y(require)14 b(is)g(its)f(mo)q(di\014ed)f(cycle)j (index.)j(Accordingly)m(,)12 b(for)i(an)o(y)f(oligomo)o(rphic)e(group)j Fn(K)s Ft(,)262 2160 y(w)o(e)g(can)g(de\014ne)h(an)e(op)q(erator)i Fn(K)i Ft(on)c(sequences)k(b)o(y)d(using)f(this)h(rule,)g(so)g(that)726 2251 y Fn(K)s Ft(\()p Fn(f)800 2257 y Fj(n)823 2251 y Ft(\()p Fn(H)s Ft(\)\))e(=)g(\()p Fn(f)1001 2257 y Fj(n)1024 2251 y Ft(\()p Fn(H)e Ft(W)m(r)c Fn(K)s Ft(\)\))p Fn(:)324 2343 y Ft(In)k(a)g(similar)d(w)o(a)o(y)m(,)j(wreath)g(pro)q(ducts)i (de\014ne)f(op)q(erators)g(on)f(the)h(sequences)h(\()p Fn(F)1573 2349 y Fj(n)1596 2343 y Ft(\()p Fn(H)s Ft(\)\).)262 2392 y(These)20 b(op)q(erators)h(are)f(m)o(uc)o(h)e(easier)j(to)e(w)o (ork)g(with,)h(since)h(they)f(are)g(just)g(giv)o(en)f(b)o(y)957 2574 y(18)p eop %%Page: 19 19 19 18 bop 262 307 a Ft(substitution)18 b(in)g(the)i(exp)q(onen)o(tial)e (generating)h(functions,)g(after)g(\014rst)g(remo)o(ving)e(the)262 357 y(constan)o(t)d(term:)715 407 y Fn(F)742 413 y Fj(H)8 b Fl(W)m(r)e Fj(K)858 407 y Ft(\()p Fn(t)p Ft(\))12 b(=)g Fn(F)988 413 y Fj(K)1019 407 y Ft(\()p Fn(F)1062 413 y Fj(H)1094 407 y Ft(\()p Fn(t)p Ft(\))d Fm(\000)h Ft(1\))p Fn(:)324 482 y Ft(The)19 b(most)f(famous)f(case)i(of)g(this)f(o)q (ccurs)j(when)e Fn(H)j Ft(is)c(the)i(symmetric)d(group)h Fn(S)r Ft(.)262 531 y(W)m(e)i(ha)o(v)o(e)h Fn(F)470 537 y Fj(S)493 531 y Ft(\()p Fn(t)p Ft(\))j(=)f(exp\()p Fn(t)p Ft(\),)g(and)d Fn(F)878 537 y Fj(S)7 b Fl(W)m(r)g Fj(K)987 531 y Ft(\()p Fn(t)p Ft(\))23 b(=)g Fn(F)1139 537 y Fj(K)1171 531 y Ft(\(exp)q(\()p Fn(t)p Ft(\))14 b Fm(\000)g Ft(1\).)39 b(In)20 b(particular,)262 581 y Fn(F)289 587 y Fj(S)7 b Fl(W)m(r)f Fj(S)389 581 y Ft(\()p Fn(t)p Ft(\))20 b(=)f(exp)q (\(exp\()p Fn(t)p Ft(\))13 b Fm(\000)f Ft(1\),)20 b(the)f(exp)q(onen)o (tial)f(generating)h(function)f(for)g(the)h(se-)262 631 y(quence)f(of)e Fo(Bel)r(l)h(numb)n(ers)p Ft(.)26 b(\(The)18 b Fn(n)p Ft(th)f(Bell)f(n)o(um)o(b)q(er)g(coun)o(ts)h(partitions)g(of)f (an)g Fn(n)p Ft(-set,)262 681 y(that)h(is,)g(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)17 b(structures)i(for)e(the)g(group)g Fn(S)10 b Ft(W)m(r)c Fn(S)r Ft(.\))28 b(This)17 b(op)q(eration)g(has)g (another)262 731 y(in)o(terpretation.)22 b(If)15 b Fn(F)622 716 y Fk(\003)616 741 y Fj(n)641 731 y Ft(\()p Fn(G)p Ft(\))g(denotes)i(the)f(n)o(um)o(b)q(er)e(of)h(orbits)g(of)g Fn(G)g Ft(on)g(all)f Fn(n)p Ft(-tuples)i(\(of)262 780 y(not)d(necessarily)i(distinct)f(elemen)o(ts\),)g(then)g(w)o(e)h(ha)o (v)o(e)780 872 y Fn(F)813 855 y Fk(\003)807 882 y Fj(n)832 872 y Ft(\()p Fn(G)p Ft(\))c(=)h Fn(F)979 878 y Fj(n)1001 872 y Ft(\()p Fn(S)e Ft(W)m(r)d Fn(G)p Ft(\))p Fn(;)262 963 y Ft(as)15 b(can)i(b)q(e)f(seen)h(b)o(y)f(replacing)g(iden)o(tical) f(p)q(oin)o(ts)h(of)f(\001)g(in)h(an)f Fn(n)p Ft(-tuple)h(\(where)h Fn(G)f Ft(acts)262 1013 y(on)c(\001\))h(b)o(y)g(distinct)g(p)q(oin)o (ts)g(of)f(the)i(\014bre)f(o)o(v)o(er)g(that)g(p)q(oin)o(t.)18 b(F)m(urthermore,)12 b(this)h(relation)262 1063 y(is)g(equiv)n(alen)o (t)g(to)733 1143 y Fn(F)766 1126 y Fk(\003)760 1154 y Fj(n)785 1143 y Ft(\()p Fn(G)p Ft(\))e(=)925 1092 y Fj(n)905 1104 y Fg(X)905 1193 y Fj(k)q Fl(=1)972 1143 y Fn(S)r Ft(\()p Fn(n;)c(k)q Ft(\))p Fn(F)1125 1149 y Fj(k)1146 1143 y Ft(\()p Fn(G)p Ft(\))p Fn(;)262 1258 y Ft(where)12 b Fn(S)r Ft(\()p Fn(n;)7 b(k)q Ft(\))k(is)g(the)h Fo(Stirling)g(numb)n (er)g(of)g(the)h(se)n(c)n(ond)g(kind)p Ft(,)e(the)h(n)o(um)o(b)q(er)e (of)h(partitions)262 1308 y(of)h(an)h Fn(n)p Ft(-set)g(in)o(to)f Fn(k)i Ft(parts.)19 b(The)13 b(op)q(erator)g(on)g(sequences)j(giv)o(en) c(b)o(y)h(the)g(ab)q(o)o(v)o(e)g(form)o(ula)262 1357 y(is)g(called)h(STIRLING)f(b)o(y)h(Bernstein)h(and)f(Sloane)f([3].)324 1407 y(\\Dual")e(to)j(this)f(op)q(erator,)g(in)g(some)f(sense,)j(is)e (the)g(op)q(erator)h(whic)o(h)f(maps)f(\()p Fn(F)1590 1413 y Fj(n)1613 1407 y Ft(\()p Fn(G)p Ft(\)\))262 1457 y(to)h(\()p Fn(F)355 1463 y Fj(n)378 1457 y Ft(\()p Fn(G)7 b Ft(W)m(r)f Fn(S)r Ft(\)\),)14 b(giv)o(en)f(b)o(y)h Fn(F)774 1463 y Fj(G)5 b Fl(W)m(r)i Fj(S)879 1457 y Ft(\()p Fn(t)p Ft(\))12 b(=)f(exp)q(\()p Fn(F)1088 1463 y Fj(G)1116 1457 y Ft(\()p Fn(t)p Ft(\))e Fm(\000)g Ft(1\).)18 b(This)c(op)q (erator,)g(referred)262 1507 y(to)f(as)g(EXP)h(in)e([3],)g(maps)g(the)i (sequence)h(en)o(umerating)d(lab)q(elled)h(connected)i(structures)262 1557 y(in)f(some)g(class)h(to)g(arbitrary)g(lab)q(elled)f(structures)k (in)c(the)i(class;)f(the)g(same)f(job)h(that)g Fn(S)262 1606 y Ft(\(or)i(EULER\))g(do)q(es)h(for)f(the)h(unlab)q(elled)f (structures.)30 b(Explicitly)m(,)16 b(it)h(is)g(giv)o(en)g(b)o(y)g(the) 262 1656 y(recurrence)731 1729 y Fn(A)762 1735 y Fj(n)796 1729 y Ft(=)860 1677 y Fj(n)840 1689 y Fg(X)840 1779 y Fj(k)q Fl(=1)907 1670 y Fg(\022)938 1701 y Fn(n)9 b Fm(\000)g Ft(1)939 1757 y Fn(k)h Fm(\000)f Ft(1)1034 1670 y Fg(\023)1065 1729 y Fn(C)1095 1735 y Fj(k)1115 1729 y Fn(A)1146 1735 y Fj(n)p Fk(\000)p Fj(k)1213 1729 y Fn(;)262 1843 y Ft(where)21 b(\()p Fn(C)434 1849 y Fj(n)456 1843 y Ft(\))h(=)h(\()p Fn(F)592 1849 y Fj(n)614 1843 y Ft(\()p Fn(G)p Ft(\)\))d(coun)o(ts)h(connected)h(ob)r(jects)g (and)e(\()p Fn(A)1335 1849 y Fj(n)1358 1843 y Ft(\))i(=)g(\()p Fn(F)1493 1849 y Fj(n)1515 1843 y Ft(\()p Fn(G)7 b Ft(W)m(r)g Fn(S)r Ft(\)\))262 1893 y(coun)o(ts)14 b(arbitrary)g(ones.)262 2030 y Fp(9)66 b(A)23 b(pro)r(duct)g(iden)n(tit)n(y)262 2121 y Ft(This)13 b(section)i(con)o(tains)f(a)f(pro)q(of)h(of)f(the)i (iden)o(tit)o(y)711 2243 y(e)729 2226 y Fj(t=)p Fl(\(1)p Fk(\000)p Fj(t)p Fl(\))853 2243 y Ft(=)912 2192 y Fk(1)902 2204 y Fg(Y)897 2292 y Fj(n)p Fl(=1)960 2243 y Ft(\(1)9 b Fm(\000)g Fn(t)1062 2226 y Fj(n)1085 2243 y Ft(\))1101 2226 y Fk(\000)p Fj(\036)p Fl(\()p Fj(n)p Fl(\))p Fj(=n)1233 2243 y Fn(;)262 2371 y Ft(where)i Fn(\036)f Ft(is)g(Euler's)g(totien)o (t)g(function.)17 b(W)m(e)10 b(need)h(another)f(example)f(of)h(an)g (oligomo)o(rphic)262 2420 y(group.)957 2574 y(19)p eop %%Page: 20 20 20 19 bop 324 307 a Ft(Let)17 b Fn(C)i Ft(b)q(e)e(the)g(group)g(of)f (all)f(p)q(erm)o(utations)h(preserving)h(the)h(cyclic)e(order)i(on)e (the)262 357 y(complex)10 b(ro)q(ots)i(of)g(unit)o(y)m(.)k(\(The)d (cyclic)f(order)g(is)g(a)g(ternary)g(relation)g Fn(R)g Ft(whic)o(h)f(holds)h(for)262 407 y(\()p Fn(x;)7 b(y)q(;)g(z)r Ft(\))k(when)h(the)h(p)q(oin)o(ts)e(are)i(visited)e(in)h(this)f(order)i (starting)f(at)f Fn(x)h Ft(and)f(pro)q(ceeding)i(in)262 457 y(an)i(an)o(ticlo)q(c)o(kwise)g(sense)i(around)f(the)g(circle;)g (so,)g(if)e Fn(R)p Ft(\()p Fn(x;)7 b(y)q(;)g(z)r Ft(\))16 b(holds,)f(then)h Fn(R)p Ft(\()p Fn(y)q(;)7 b(z)r(;)g(x)p Ft(\))262 506 y(holds)16 b(but)h Fn(R)p Ft(\()p Fn(x;)7 b(z)r(;)g(y)q Ft(\))16 b(do)q(esn't.\))27 b(The)17 b(group)g Fn(C)i Ft(is)e(transitiv)o(e,)g(and)f(the)i(stabiliser)e(of)262 556 y(a)i(p)q(oin)o(t)g(preserv)o(es)k(a)c(linear)h(order)g(on)g(the)g (remaining)e(p)q(oin)o(ts;)k(so)e(the)g(stabiliser)g(is)262 606 y(isomorphic)12 b(to)i Fn(A)p Ft(.)k(Using)13 b(this)h(fact,)g(or)g (b)o(y)f(sho)o(wing)h(that)g(the)g(relational)f(structure)j(is)262 656 y(homogeneous)e(\(m)o(uc)o(h)g(as)h(w)o(e)h(did)e(for)h Fn(A)g Ft(earlier\),)h(w)o(e)f(see)i(that)e Fn(C)j Ft(has)d(just)g(one) h(orbit)262 706 y(on)d Fn(n)p Ft(-sets)i(for)f(ev)o(ery)g Fn(n)e(>)f Ft(0,)j(and)f(the)i(stabiliser)f(of)f(an)g Fn(n)p Ft(-set)i(induces)g(on)e(it)h(the)g(cyclic)262 756 y(group)f Fn(C)409 762 y Fj(n)445 756 y Ft(of)h(order)g Fn(n)p Ft(.)324 805 y(No)o(w)g Fn(C)449 811 y Fj(n)485 805 y Ft(con)o(tains)h Fn(\036)p Ft(\()p Fn(d)p Ft(\))f(elemen)o(ts)g (of)g(order)h Fn(d)f Ft(for)g(eac)o(h)g(divisor)g Fn(d)g Ft(of)g Fn(n)p Ft(;)g(and)g(eac)o(h)262 855 y(of)f(these)i(elemen)o(ts) f(has)g Fn(n=d)f Ft(cycles)i(of)e(length)h Fn(d)p Ft(.)k(So)13 b(w)o(e)h(ha)o(v)o(e)660 968 y(~)651 979 y Fn(Z)s Ft(\()p Fn(C)s Ft(\))42 b(=)g(1)9 b(+)949 927 y Fk(1)935 939 y Fg(X)934 1027 y Fj(n)p Fl(=1)1011 951 y Ft(1)p 1008 969 25 2 v 1008 1007 a Fn(n)1045 939 y Fg(X)1051 1030 y Fj(d)p Fk(j)p Fj(n)1112 979 y Fn(\036)p Ft(\()p Fn(d)p Ft(\))p Fn(s)1210 957 y Fj(n=d)1210 991 y(d)789 1127 y Ft(=)42 b(1)9 b(+)947 1075 y Fk(1)934 1088 y Fg(X)934 1177 y Fj(d)p Fl(=1)1006 1099 y Fn(\036)p Ft(\()p Fn(d)p Ft(\))p 1006 1117 79 2 v 1034 1155 a Fn(d)1116 1075 y Fk(1)1102 1088 y Fg(X)1096 1175 y Fj(m)p Fl(=1)1180 1099 y Fn(s)1199 1084 y Fj(m)1199 1111 y(d)p 1180 1117 51 2 v 1187 1155 a Fn(m)789 1266 y Ft(=)42 b(1)9 b Fm(\000)947 1214 y Fk(1)934 1227 y Fg(X)934 1316 y Fj(d)p Fl(=1)1006 1238 y Fn(\036)p Ft(\()p Fn(d)p Ft(\))p 1006 1257 79 2 v 1034 1295 a Fn(d)1096 1266 y Ft(log\(1)g Fm(\000)h Fn(s)1257 1272 y Fj(d)1276 1266 y Ft(\))p Fn(:)324 1427 y Ft(Since)k Fn(f)452 1433 y Fj(n)475 1427 y Ft(\()p Fn(C)s Ft(\))e(=)f(1)j(for)f(all)g Fn(n)p Ft(,)g(w)o(e)i(ha)o(v)o(e)e Fn(f)978 1433 y Fj(C)1006 1427 y Ft(\()p Fn(t)p Ft(\))f(=)g(1)p Fn(=)p Ft(\(1)d Fm(\000)g Fn(t)p Ft(\))j(=)g(1)d(+)g Fn(t=)p Ft(\(1)g Fm(\000)h Fn(t)p Ft(\).)18 b(Hence)618 1551 y(1)9 b(+)731 1523 y Fn(t)p 695 1542 87 2 v 695 1580 a Ft(1)g Fm(\000)g Fn(t)798 1551 y Ft(=)j(1)d Fm(\000)927 1499 y Fk(1)913 1512 y Fg(X)914 1601 y Fj(d)p Fl(=1)973 1551 y Ft(\()p Fn(\036)p Ft(\()p Fn(d)p Ft(\))p Fn(=d)p Ft(\))e(log)o(\(1)i Fm(\000)h Fn(t)1290 1534 y Fj(d)1309 1551 y Ft(\))p Fn(:)262 1679 y Ft(No)o(w)17 b(subtracting)i(1)f(from)e (eac)o(h)i(side,)h(taking)e(the)i(exp)q(onen)o(tial,)f(and)g(replacing) g(the)262 1729 y(dumm)o(y)10 b(v)n(ariable)j Fn(d)h Ft(b)o(y)f Fn(n)h Ft(giv)o(es)g(the)g(result.)324 1778 y(Note)c(that,)f(ha)o(ving) g(w)o(ork)o(ed)g(out)862 1768 y(~)853 1778 y Fn(Z)s Ft(\()p Fn(C)s Ft(\),)h(w)o(e)g(can)f(write)h(do)o(wn)f(the)h(sequence)h(op)q (erator)262 1828 y(corresp)q(onding)j(to)g Fn(C)s Ft(,)f(in)h(terms)f (of)g(its)h(action)g(on)g(generating)g(functions:)630 1950 y(\()p Fn(C)s(f)t Ft(\)\()p Fn(t)p Ft(\))e(=)g(1)d Fm(\000)908 1898 y Fk(1)895 1910 y Fg(X)894 1998 y Fj(n)p Fl(=1)968 1922 y Fn(\036)p Ft(\()p Fn(n)p Ft(\))p 968 1940 82 2 v 997 1978 a Fn(n)1062 1950 y Ft(log)o(\(2)h Fm(\000)f Fn(f)t Ft(\()p Fn(t)1258 1933 y Fj(n)1282 1950 y Ft(\)\))p Fn(:)324 2100 y Ft(Ha)o(ving)17 b(added)i Fn(C)i Ft(to)e(our)f(rep)q(ertoire,)j(it)d(is)h(in)o(teresting)g(to)f (consider)i(the)f(group)262 2150 y Fn(C)9 b Ft(W)m(r)d Fn(S)r Ft(.)21 b(A)15 b(mem)o(b)q(er)e(of)h(the)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)g(for)f(it)g(consists)i(of)e (a)g(set)h(carrying)g(a)f(parti-)262 2200 y(tion)g(with)h(a)g(circular) g(order)h(on)f(eac)o(h)g(part.)22 b(This)15 b(is)g(precisely)h(the)g (sp)q(eci\014cation)f(of)g(a)262 2250 y(p)q(erm)o(utation,)9 b(decomp)q(osed)h(in)o(to)g(disjoin)o(t)f(cycles.)18 b(So)10 b(the)g(group)g Fn(C)g Ft(W)m(r)c Fn(S)13 b Ft(\\represen)o (ts")262 2300 y(p)q(erm)o(utations.)324 2350 Y(G)(O)(O)(Q)(f)(g)(p)q(ERM)o(UTION)f(and)i(of)E(总)H(阶)i(on)e(an)g fn(n)p f-(-set)h(是)g(b)q(Ot)g(等)262 2399 y(to)19πfn(n)p FT(!).)36 b(So)19 b(there)i(should)f(b)q(e)h(some)e(relation)g (b)q(et)o(w)o(een)i Fn(C)10 b Ft(W)m(r)c Fn(S)23 b Ft(and)d Fn(A)p Ft(.)36 b(Ho)o(w)o(ev)o(er,)262 2449 y(the)18 b(bijection)g(b)q(et)o(w)o(een)h(linear)f(orders)h(and)e(p)q(erm)o (utations)g(is)h(not)g(a)g(\\natural")f(one:)957 2574 y(20)p eop %%Page: 21 21 21 20 bop 262 307 a Ft(w)o(e)19 b(m)o(ust)f(\014rst)i(c)o(ho)q(ose)f(a) g(distinguished)g(order)h Fn(\025)p Ft(,)g(and)e(then)i(an)o(y)f(other) g(order)h(is)f(a)262 357 y(p)q(erm)o(utation)12 b(of)h Fn(\025)p Ft(.)324 407 y(W)m(e)k(kno)o(w)f(already)h(that)762 396 y(~)753 407 y Fn(Z)t Ft(\()p Fn(A)p Ft(\))g(=)h(1)p Fn(=)p Ft(\(1)10 b Fm(\000)i Fn(s)1067 413 y Fl(1)1086 407 y Ft(\).)28 b(A)17 b(straigh)o(tforw)o(ard)g(calculation,)262 457 y(using)11 b(the)i(v)n(alue)f(of)598 446 y(~)589 457 y Fn(Z)s Ft(\()p Fn(C)s Ft(\))h(found)e(ab)q(o)o(v)o(e,)h(sho)o(ws) g(that)1154 446 y(~)1146 457 y Fn(Z)s Ft(\()p Fn(C)e Ft(W)m(r)c Fn(S)r Ft(\))13 b(=)1394 425 y Fg(Q)1434 469 y Fj(n)p Fk(\025)p Fl(1)1499 457 y Ft(\(1)6 b Fm(\000)g Fn(s)1599 463 y Fj(n)1621 457 y Ft(\))1637 442 y Fk(\000)p Fl(1)1682 457 y Ft(.)262 506 y(These)16 b(t)o(w)o(o)e(expressions)j (are)e(di\013eren)o(t;)i(but,)e(to)f(compute)h(the)h(e.g.f.)k(for)15 b(the)g(n)o(um)o(b)q(er)262 556 y(of)g(lab)q(elled)h(structures,)j(w)o (e)d(substitute)i Fn(t)e Ft(for)g Fn(s)1052 562 y Fl(1)1087 556 y Ft(and)g(0)g(for)g Fn(s)1292 562 y Fj(n)1331 556 y Ft(\()p Fn(n)g(>)g Ft(1\);)h(the)f(results)262 606 y(are)e(the)g(same,)f(as)h(they)g(should)g(b)q(e:)695 697 y Fn(F)722 703 y Fj(A)749 697 y Ft(\()p Fn(t)p Ft(\))d(=)h Fn(F)878 703 y Fj(C)7 b Fl(W)m(r)g Fj(S)983 697 y Ft(\()p Fn(t)p Ft(\))12 b(=)f(\(1)f Fm(\000)f Fn(t)p Ft(\))1204 680 y Fk(\000)p Fl(1)1249 697 y Fn(:)262 835 y Fp(10)66 b(Stirling)25 b(n)n(um)n(b)r(ers)262 926 y Ft(W)m(e)10 b(already)h(sa)o(w)g(that)g(Stirling)f(n)o(um)o(b)q(ers)h(are)g(in)o(v) o(olv)o(ed)f(with)h(the)g(formalism)d(of)i(wreath)262 975 y(pro)q(ducts.)19 b(It)14 b(is)g(p)q(ossible)g(to)f(de\014ne)i(and) f(generalise)g(them)g(using)f(this)h(philosoph)o(y)m(.)324 1025 y(I)g(b)q(egin)g(with)g(a)g(brief)g(course)h(on)f(Stirling)f(n)o (um)o(b)q(ers.)18 b(The)d Fo(Stirling)f(numb)n(er)i(of)f(the)262 1075 y(\014rst)g(kind)p Ft(,)g Fn(S)r Ft(\()p Fn(n;)7 b(k)q Ft(\),)14 b(is)h(the)g(n)o(um)o(b)q(er)f(of)g(partitions)h(of)f (an)g Fn(n)p Ft(-set)h(in)o(to)f Fn(k)i Ft(parts.)21 b(W)m(e)14 b(see)262 1125 y(imm)o(ediately)e(that)j(the)g(sum)751 1094 y Fg(P)794 1104 y Fj(n)794 1137 y(k)q Fl(=1)864 1125 y Fn(S)r Ft(\()p Fn(n;)7 b(k)q Ft(\))13 b(=)g Fn(B)r Ft(\()p Fn(n)p Ft(\))j(\(the)f Fo(Bel)r(l)g(numb)n(er)p Ft(\))g(is)g(the)g(total)262 1175 y(n)o(um)o(b)q(er)e(of)g(partitions)g (of)h(an)f Fn(n)p Ft(-set,)h(whic)o(h)g(w)o(e)g(recognise)h(as)f Fn(F)1305 1181 y Fj(n)1327 1175 y Ft(\()p Fn(S)c Ft(W)m(r)d Fn(S)r Ft(\).)324 1224 y(The)16 b Fo(unsigne)n(d)i(Stirling)e(numb)n (er)h(of)g(the)g(se)n(c)n(ond)g(kind)p Ft(,)f Fn(s)p Ft(\()p Fn(n;)7 b(k)q Ft(\),)16 b(is)g(the)h(n)o(um)o(b)q(er)e(of)262 1274 y(p)q(erm)o(utations)f(of)g(an)h Fn(n)p Ft(-set)h(with)f Fn(k)h Ft(disjoin)o(t)e(cycles.)23 b(Th)o(us)15 b(w)o(e)h(ha)o(v)o(e) 1417 1243 y Fg(P)1460 1253 y Fj(n)1460 1287 y(k)q Fl(=1)1530 1274 y Fn(s)p Ft(\()p Fn(n;)7 b(k)q Ft(\))13 b(=)262 1324 y Fn(n)p Ft(!)e(=)i Fn(F)382 1330 y Fj(n)404 1324 y Ft(\()p Fn(A)p Ft(\).)20 b(It)14 b(is)g(more)g(useful)g(to)g(re-in)o (terpret)i(this)f(in)e(the)i(ligh)o(t)f(of)f(the)i(remarks)f(in)262 1374 y(the)e(last)f(section.)18 b(A)11 b(p)q(erm)o(utation)g(with)g Fn(k)h Ft(cycles)h(is)e(giv)o(en)g(b)o(y)g(a)g(partition)g(in)o(to)g Fn(k)h Ft(parts)262 1424 y(with)h(a)h(cyclic)g(order)g(on)g(eac)o(h)g (part;)g(and)g(w)o(e)g(ha)o(v)o(e)1104 1393 y Fg(P)1147 1403 y Fj(n)1147 1436 y(k)q Fl(=1)1217 1424 y Fn(s)p Ft(\()p Fn(n;)7 b(k)q Ft(\))k(=)h Fn(F)1417 1430 y Fj(n)1439 1424 y Ft(\()p Fn(C)e Ft(W)m(r)d Fn(S)r Ft(\).)324 1474 y(This)20 b(immediately)d(suggests)22 b(a)f(generalisation.)37 b(Let)21 b Fn(G)f Ft(b)q(e)h(an)o(y)g(oligomo)o(rphic)262 1523 y(p)q(erm)o(utation)d(group.)34 b(W)m(e)19 b(de\014ne)h(the)g Fo(gener)n(alise)n(d)g(Stirling)f(numb)n(er)g Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))19 b(to)262 1573 y(b)q(e)g(the)g(n)o(um)o(b)q(er)g(of)f(partitions)g(of)g(an)h Fn(n)p Ft(-set)h(in)o(to)e Fn(k)h Ft(parts,)i(with)d(a)h(mem)o(b)q(er)e (of)h(the)262 1623 y(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)13 b(class)h(for)f Fn(G)g Ft(on)g(eac)o(h)h(part.)k(Th)o(us)13 b(w)o(e)h(ha)o(v)o(e)1120 1592 y Fg(P)1164 1602 y Fj(n)1164 1635 y(k)q Fl(=1)1233 1623 y Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))k(=)h Fn(F)1498 1629 y Fj(n)1520 1623 y Ft(\()p Fn(G)7 b Ft(W)m(r)f Fn(S)r Ft(\).)262 1673 y(In)15 b(this)h(notation,)f(the)i(\\classical")e(Stirling)g(n)o (um)o(b)q(ers)g(are)h Fn(S)r Ft(\()p Fn(n;)7 b(k)q Ft(\))16 b(=)f Fn(S)r Ft([)p Fn(S)r Ft(]\()p Fn(n;)7 b(k)q Ft(\))16 b(and)262 1723 y Fn(s)p Ft(\()p Fn(n;)7 b(k)q Ft(\))k(=)h Fn(S)r Ft([)p Fn(C)s Ft(]\()p Fn(n;)7 b(k)q Ft(\).)324 1772 y(It)16 b(is)h(clear)g(that)f(the)h(generalised)h(Stirling)d(n)o (um)o(b)q(ers)h Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))16 b(are)h(determined)262 1822 y(b)o(y)e(the)i(n)o(um)o(b)q (ers)f Fn(F)592 1828 y Fj(n)614 1822 y Ft(\()p Fn(G)p Ft(\).)24 b(This)16 b(can)h(b)q(e)f(expressed)j(most)c(concisely)h(in)g (terms)g(of)f(the)262 1872 y(exp)q(onen)o(tial)e(generating)h (functions:)645 1942 y Fk(1)632 1955 y Fg(X)629 2044 y Fj(n)p Fl(=)p Fj(k)701 1994 y Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))p Fn(t)899 1977 y Fj(n)920 1994 y Fn(=n)p Ft(!)k(=)fn(p)fn(f)1076 y y fJ(g)1104π1994 y FT(\(p)fn(t)p ft(\))d Fm(\ 000)h FT(1))1239 1977μy fJ(k)1259 1994 y fn(=k)q FT(?)p Fn(:)262 2125 y Ft(F)m(rom)15 b(this,)i(the)g(equation)g Fn(F)744 2131 y Fj(G)5 b Fl(W)m(r)h Fj(S)848 2125 y Ft(\()p Fn(t)p Ft(\))17 b(=)g(exp\()p Fn(F)1067 2131 y Fj(G)1095 2125 y Ft(\()p Fn(t)p Ft(\))12 b Fm(\000)f Ft(1\))17 b(is)f(obtained)h(b)o(y)g(summing)262 2175 y(o)o(v)o(er)c Fn(k)q Ft(.)324 2250 y(The)h(generalised)h (Stirling)d(n)o(um)o(b)q(ers)i(ha)o(v)o(e)g(a)f(comp)q(osition)f(prop)q (ert)o(y:)612 2320 y Fj(n)592 2333 y Fg(X)595 2422 y Fj(l)p Fl(=)p Fj(k)659 2372 y Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(l)q Ft(\))p Fn(S)r Ft([)p Fn(H)s Ft(]\()p Fn(l)q(;)g(k)q Ft(\))j(=)i Fn(S)r Ft([)p Fn(G)7 b Ft(W)m(r)g Fn(H)s Ft(]\()p Fn(n;)g(k)q Ft(\))p Fn(:)957 2574 y Ft(21)p eop %%Page: 22 22 22 21 bop 262 307 a Ft(F)m(or)18 b(consider)i Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(l)q Ft(\))p Fn(S)r Ft([)p Fn(H)s Ft(]\()p Fn(l)q(;)g(k)q Ft(\).)33 b(This)19 b(coun)o(ts)h(pairs) f(consisting)g(of)g(a)f(partition)262 357 y(of)e Fm(f)p Ft(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(n)p Fm(g)16 b Ft(in)o(to)h Fn(l)h Ft(parts)g(with)f(a)f Fn(G)p Ft(-structure)j(on)e(eac)o(h)h (part,)f(and)g(a)g(partition)f(of)262 407 y(the)k(set)h(of)e(parts)h (in)o(to)f Fn(k)i Ft(parts)f(with)f(an)h Fn(H)s Ft(-structure)i(on)d (eac)o(h)h(part.)36 b(\(Here)21 b(\\)p Fn(G)p Ft(-)262 457 y(structure")16 b(is)f(short)g(for)g(\\mem)o(b)q(er)e(of)h(the)i(F) m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)15 b(class)g(of)f Fn(G)p Ft(".\))21 b(View)o(ed)15 b(otherwise,)262 506 y(w)o(e)c(ha)o(v)o(e)g(a)g(partition)g(of)f Fm(f)p Ft(1)p Fn(;)d(:)g(:)g(:)e(;)i(n)p Fm(g)j Ft(in)o(to)g Fn(k)j Ft(parts,)e(eac)o(h)h(part)g(carrying)f(a)g(partition)f(in)o(to)262 556 y(\\subparts")15 b(with)g(a)g Fn(G)p Ft(-structure)i(on)e(eac)o(h)h (subpart)g(and)f(an)g Fn(H)s Ft(-structure)i(on)e(the)h(set)262 606 y(of)c(subparts)i(\(in)f(other)h(w)o(ords,)f(a)g Fn(G)7 b Ft(W)m(r)f Fn(H)s Ft(-structure\),)15 b(sub)r(ject)g(to)e(the) h(condition)f(that)262 656 y(there)g(are)f Fn(l)h Ft(subparts)g (altogether.)k(Summing)9 b(o)o(v)o(er)j Fn(l)h Ft(remo)o(v)o(es)e(the)h (\014nal)g(condition)f(and)262 706 y(yields)i Fn(S)r Ft([)p Fn(G)7 b Ft(W)m(r)g Fn(H)s Ft(]\()p Fn(n;)g(k)q Ft(\).)324 756 y(This)j(result)g(can)h(b)q(e)f(expressed)i(more)d (compactly)g(in)g(matrix)f(form.)15 b(Let)c Fn(T)6 b Ft([)p Fn(G)p Ft(])i(b)q(e)j(the)262 805 y(triangular)i(arra)o(y)h(of)g (generalised)h(Stirling)f(n)o(um)o(b)q(ers)g(asso)q(ciated)h(with)f Fn(G)p Ft(,)g(the)h(in\014nite)262 855 y(lo)o(w)o(er)e(triangular)g(m)o (trix)f(with)i(\()p Fn(n;)7 b(k)q Ft(\))13 b(en)o(try)i Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\).)17 b(Then)d(w)o(e)g(ha)o(v)o(e)759 945 y Fn(T)6 b Ft([)p Fn(G)p Ft(])p Fn(T)g Ft([)p Fn(H)s Ft(])k(=)h Fn(T)6 b Ft([)p Fn(G)h Ft(W)m(r)f Fn(H)s Ft(])p Fn(:)262 1036 y Ft(F)m(or)11 b(example,)g Fn(T)6 b Ft([)p Fn(S)r Ft(])13 b(and)f Fn(T)6 b Ft([)p Fn(C)s Ft(])11 b(are)i(the)g(arra)o(ys)f(of)g (classical)g(Stirling)f(n)o(um)o(b)q(ers;)h(and)g(w)o(e)262 1086 y(ha)o(v)o(e)700 1135 y Fn(T)6 b Ft([)p Fn(C)s Ft(])p Fn(T)g Ft([)p Fn(S)r Ft(])k(=)i Fn(T)6 b Ft([)p Fn(C)j Ft(W)m(r)d Fn(S)r Ft(])12 b(=)g Fn(T)6 b Ft([)p Fn(A)p Ft(])p Fn(:)324 1210 y Ft(The)23 b(n)o(um)o(b)q(ers)f Fn(S)r Ft([)p Fn(A)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))22 b(are)h(the)g Fo(L)n(ah)g(numb)n(ers)g Fn(L)p Ft(\()p Fn(n;)7 b(k)q Ft(\),)24 b(sometimes)d(called)262 1259 y(\\Stirling)14 b(n)o(um)o(b)q(ers)h(of)h(the)g(third)g(kind":)21 b(see)c(Lah)e([17)o(],)h(Bridgeman)e([5].)23 b(Unlik)o(e)15 b(the)262 1309 y(classical)e(Stirling)g(n)o(um)o(b)q(ers,)g(there)i(is) f(a)f(closed)i(form)o(ula)c(for)j(the)g(Lah)g(n)o(um)o(b)q(ers:)633 1422 y Fn(L)p Ft(\()p Fn(n;)7 b(k)q Ft(\))k(=)820 1394 y(\()p Fn(n)e Fm(\000)h Ft(1\)!)p 820 1413,141 2 V 821 1451 A(\)p fn(k)g Fm(\000)g FT(1)!965 1364 y fg(022)996 y 1394 y fn(n)997 1451 y(k)1021 1364 y fg(023)1063 1422 y y(=)1422 y fn(n)p ft(!)p 1112 1413 37 37 V 1113 1451 1451 FN(k)q FT(?)1153 1364 y Fg(\022)1184 1394 y Fn(n)f Fm(\000)g Ft(1)1185 1451 y Fn(k)h Fm(\000)f Ft(1)1280 1364 y Fg(\023)1311 1422 y Fn(:)262 1535 y Ft(This)k(can)h(b)q(e)h(sho) o(wn)f(b)o(y)f(using)h(the)h(form)o(ula)686 1618 y Fg(X)683 1708 y Fj(n)p Fk(\025)p Fj(k)755 1658 y Fn(L)p Ft(\()p Fn(n;)7 b(k)q Ft(\))p Fn(t)897 1641 y Fj(n)919 1658 y Fn(=n)p Ft(!)K(=1032)1599 y fg(022)1103 y y fn(t)p 1067 1067 1648 87 2 v 1067 1686 a英尺(1)e fm(\000)h fn(t)αy y fg(α)y fJ(k)y y fn(=k)q FT(?)262 1791 y(and)j(computing)f(the)j(co)q(e\016cien)o(t)g(of)e Fn(t)875 1776 y Fj(n)911 1791 y Ft(on)h(the)h(righ)o(t-hand)e(side.)324 1865 y(In)h(a)f(similar)f(manner,)g(it)i(can)g(b)q(e)g(sho)o(wn)g(that) 670 1935 y Fj(n)650 1947 y Fg(X)650 2036 y Fj(k)q Fl(=1)717 1987 y Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))p Fn(F)927 1993 y Fj(k)947 1987 y Ft(\()p Fn(H)s Ft(\))k(=)h Fn(F)1099 1993 y Fj(n)1121 1987 y Ft(\()p Fn(G)7 b Ft(W)m(r)g Fn(H)s Ft(\))p Fn(:)262 2114 y Ft(This)13 b(prop)q(ert)o(y)i (generalises)g(the)f(STIRLING)f(transform)g(w)o(e)h(met)f(earlier.)324 2188 y(There)h(is)e(another)h(remark)n(able)f(prop)q(ert)o(y)h(of)f (classical)h(Stirling)e(and)i(Lah)f(n)o(um)o(b)q(ers.)262 2238 y(Let)e Fn(S)359 2223 y Fk(\003)379 2238 y Ft([)p Fn(G)p Ft(]\()p Fn(n;)d(k)q Ft(\))j(=)i(\()p Fm(\000)p Ft(1\))674 2223 y Fj(n)p Fk(\000)p Fj(k)741 2238 y Fn(S)r Ft([)p Fn(G)p Ft(]\()p Fn(n;)7 b(k)q Ft(\))j(b)q(e)h(the)g Fo(signe)n(d)f Ft(generalised)h(Stirling)e(n)o(um)o(b)q(ers,)262 2288 y(and)k(let)h Fn(T)432 2273 y Fk(\003)451 2288 y Ft([)p Fn(G)p Ft(])f(b)q(e)h(the)h(corresp)q(onding)g(triangular)e (arra)o(y)m(.)k(Then)718 2360 y Fj(n)698 2372 y Fg(X)701 2462 y Fj(l)p Fl(=)p Fj(k)765 2412 y Fn(S)r Ft(\()p Fn(n;)7 b(l)q Ft(\)\()p Fm(\000)p Ft(1\))966 2395 y Fj(l)p Fk(\000)p Fj(k)1024 2412 y Fn(s)p Ft(\()p Fn(l)q(;)g(k)q Ft(\))12 b(=)g Fn(\016)1204 2418 y Fj(nk)1245 2412 y Fn(;)957 2574 y Ft(22)p eop %%Page: 23 23 23 22 bop 262 307 a Ft(or)13 b(in)h(other)g(w)o(ords)841 357 y Fn(T)6 b Ft([)p Fn(S)r Ft(])p Fn(T)952 340 y Fk(\003)971 357 y Ft([)p Fn(C)s Ft(])k(=)i Fn(I)s(:)262 432 y Ft(It)i(follo)o(ws)f (that)i(also)e Fn(T)6 b Ft([)p Fn(C)s Ft(])p Fn(T)735 417 y Fk(\003)753 432 y Ft([)p Fn(S)r Ft(])13 b(=)g Fn(I)18 b Ft(and)c Fn(T)6 b Ft([)p Fn(A)p Ft(])p Fn(T)1094 417 y Fk(\003)1112 432 y Ft([)p Fn(A)p Ft(])12 b(=)h Fn(I)s Ft(.)20 b(I)15 b(do)f(not)g(kno)o(w)g(whether)262 482 y(this)f(in)o(v)o(ersion)h(relation)f(has)h(analogues)g(for)f(other)i (groups.)262 619 y Fp(11)66 b(Stabilisers)24 b(and)f(deriv)l(ativ)n(es) 262 710 y Ft(W)m(e'v)o(e)17 b(seen)i(that)f(the)g(group-theoretic)h(op) q(erations)f(of)f(direct)i(and)e(wreath)i(pro)q(duct)262 760 y(\\corresp)q(ond")g(to)f(m)o(ultipli)o(cation)d(and)j(comp)q (osition)e(of)i(formal)d(p)q(o)o(w)o(er)k(series.)32 b(It)18 b(is)262 809 y(p)q(ossible)e(to)g(in)o(terpret)h(di\013eren)o (tiation)e(in)h(similar)d(terms.)24 b(In)16 b(这个)g(节)h(i)f(假定)262(859)y(c)h(p)q(ERm)o(UTE)O(组)h fn(g)g ft(is)g(瞬变)O(e)g(on)h(\n);)f(though)g(it)g(is)g(p)q (ossible)g(to)h(form)o(u-)262 909 y(late)g(the)i(results)g(more)e (generally)m(.)324 959 y(The)i Fo(stabiliser)e Fn(G)618 965 y Fj(\013)655 959 y Ft(of)h(the)h(p)q(oin)o(t)f Fn(\013)d Fm(2)h Ft(\012)j(is)f(the)h(subgroup)f(of)g Fn(G)g Ft(consisting)g(of)g (the)262 1009 y(p)q(erm)o(utations)g(whic)o(h)h(\014x)g Fn(\013)p Ft(.)22 b(W)m(e)15 b(consider)h(it)f(as)h(a)f(p)q(erm)o (utation)f(group)h(on)g(\012)10 b Fm(n)g(f)p Fn(\013)p Fm(g)p Ft(.)262 1058 y(No)o(w)j(w)o(e)h(ha)o(v)o(e)809 1109 y(~)800 1119 y Fn(Z)s Ft(\()p Fn(G)880 1125 y Fj(\013)904 1119 y Ft(\))d(=)999 1091 y Fn(@)p 980 1110 63 2 v 980 1148 a(@)r(s)1023 1154 y Fl(1)1056 1109 y Ft(~)1048 1119 y Fn(Z)s Ft(\()p Fn(G)p Ft(\))p Fn(:)262 1212 y Ft(It)i(follo)o(ws)g (that)808 1273 y Fn(F)835 1279 y Fj(G)861 1283 y Ff(\013)883 1273 y Ft(\()p Fn(t)p Ft(\))f(=)998 1245 y(d)p 991 1264 39 2 v 991 1302 a(d)p Fn(t)1034 1273 y(F)1061 1279 y Fj(G)1089 1273 y Ft(\()p Fn(t)p Ft(\))p Fn(:)262 1362 y Ft(\(In)g(fact,)h(it)f(is)h(easy)g(to)g(see)h(this)f(directly)m(.)k (Di\013eren)o(tiating)12 b(an)h(exp)q(onen)o(tial)f(generating)262 1412 y(function)g(corresp)q(onds)j(to)e(shifting)f(the)i(terms)f(of)f (the)i(sequence)h(one)e(place)h(to)f(the)g(left,)262 1461 y(so)g(the)i(preceding)g(equation)e(sa)o(ys)797 1553 y Fn(F)824 1559 y Fj(n)846 1553 y Ft(\()p Fn(G)895 1559 y Fj(\013)919 1553 y Ft(\))e(=)h Fn(F)1017 1559 y Fj(n)p Fl(+1)1082 1553 y Ft(\()p Fn(G)p Ft(\))p Fn(:)262 1644 y Ft(The)f(corresp)q(ondence)k(b)q(et)o(w)o(een)d(orbits)g(of)e Fn(G)982 1650 y Fj(\013)1017 1644 y Ft(on)h Fn(n)p Ft(-tuples)h(and)f (of)g Fn(G)g Ft(on)g(\()p Fn(n)t Ft(+)t(1\)-tuples)262 1694 y(can)h(b)q(e)i(describ)q(ed)g(th)o(us:)k(tak)o(e)13 b(an)g(orbit)f(of)g Fn(G)h Ft(on)f(\()p Fn(n)7 b Ft(+)g(1\)-tuples,)13 b(select)h(all)e(the)h(tuples)262 1744 y(whic)o(h)g(b)q(egin)h(with)g Fn(\013)p Ft(,)f(and)g(delete)i Fn(\013)f Ft(from)e(them.\))324 1794 y(On)i(the)g(other)h(hand,)e(the)i(sequence)h(\()p Fn(f)968 1800 y Fj(n)991 1794 y Ft(\()p Fn(G)1040 1800 y Fj(\013)1063 1794 y Ft(\)\))e(is)g(not)g(determined)g(b)o(y)g(\()p Fn(f)1536 1800 y Fj(n)1559 1794 y Ft(\()p Fn(G)p Ft(\)\).)324 1843 y(The)19 b(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)19 b(class)g(for)f Fn(G)759 1849 y Fj(\013)801 1843 y Ft(is)h(obtained)f (from)f(that)i(for)f Fn(G)g Ft(b)o(y)g(distinguishing)g(a)262 1893 y(p)q(oin)o(t)12 b Fn(x)g Ft(in)h(eac)o(h)g(\014nite)g (substructure)j(and)c(deleting)h Fn(x)p Ft(.)18 b(\(This)13 b(is)f(not)h(the)g(same)f(as)h(just)262 1943 y(deleting)h(a)g(p)q(oin)o (t,)f(since)i(it)f(lea)o(v)o(es)h(a)f(shado)o(w,)g(the)h(extra)f (structure)j(obtained)d(when)h Fn(x)262 1993 y Ft(w)o(as)g (distinguished.)24 b(F)m(or)16 b(example,)e(if)h(the)i(ob)r(jects)g(in) e(the)i(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)16 b(class)h(are)f (graphs,)262 2043 y(then)21 b(b)o(y)f(distinguishing)f(and)i(deleting)f Fn(x)g Ft(w)o(e)h(sp)q(ecify)g(a)g(subset)h(of)d(the)j(remaining)262 2092 y(v)o(ertices,)15 b(those)g(whic)o(h)f(w)o(ere)i(joined)e(to)g Fn(x)p Ft(.\))20 b(In)14 b(view)g(of)g(the)h(e\013ect)h(on)e(the)h (generating)262 2142 y(function,)e(I)g(will)g(denote)i(this)f(op)q (eration)f(on)h(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)14 b(classes)h(b)o(y)f Fn(@)r Ft(.)324 2192 y(Tw)o(o-graphs)f(pro)o(vide)h (an)g(example)e(\(see)j(Seidel)f([31)o(]\).)k(If)13 b Fn(x)h Ft(is)g(a)f(p)q(oin)o(t)h(of)f(the)h(t)o(w)o(o-)262 2242 y(graph)e(\()p Fn(X)q(;)7 b(T)f Ft(\),)13 b(there)h(is)f(a)g (unique)g(graph)g(in)f(the)i(corresp)q(onding)g(switc)o(hing)f(class)g (with)262 2292 y(the)j(prop)q(ert)o(y)g(that)g Fn(x)f Ft(is)h(an)f(isolated)h(v)o(ertex.)24 b(Th)o(us,)16 b(if)f Fn(Gr)h Ft(and)f Fn(T)6 b(w)q(oGr)16 b Ft(denote)h(the)262 2341 y(classes)e(of)e(graphs)h(and)g(t)o(w)o(o-graphs,)f(w)o(e)h(ha)o (v)o(e)840 2433 y Fn(Gr)e Ft(=)g Fn(@)r(T)6 b(w)q(oGr)o(:)957 2574 y Ft(23)p eop %%Page: 24 24 24 23 bop 324 307 a Ft(In)13 b(com)o(binatorial)e(terms,)i(it)g(is)g (more)g(natural)g(to)g(lea)o(v)o(e)g(the)h(p)q(oin)o(t)f Fn(x)g Ft(in,)g(obtaining)262 357 y(a)k(\\ro)q(oted")h(structure.)32 b(This)18 b(is)g(easily)f(handled:)26 b(adding)17 b(the)i(\014xed)f(p)q (oin)o(t)f(bac)o(k)h(in)262 407 y(corresp)q(onds)c(to)f(taking)f(the)i (direct)g(pro)q(duct)f(of)g Fn(G)1087 413 y Fj(\013)1123 407 y Ft(with)f(the)i(trivial)d(group)i(acting)g(on)262 457 y(a)g(single)h(p)q(oin)o(t,)f(whose)h(mo)q(di\014ed)f(cycle)h (index)g(is)g(1)9 b(+)g Fn(s)1170 463 y Fl(1)1189 457 y Ft(.)324 506 y(Ha)o(ving)15 b(de\014ned)j(deriv)n(ativ)o(es,)e(w)o(e) h(can)g(consider)g(di\013eren)o(tial)f(equations.)26 b(F)m(or)16 b(ex-)262 556 y(ample,)i(is)i(there)h(a)e(group)g Fn(G)g Ft(for)h(whic)o(h)f Fn(G)1004 562 y Fj(\013)1048 545 y Fm(\030)1048 558 y Ft(=)1101 556 y Fn(G)13 b Fm(\002)g Fn(G)p Ft(?)35 b(F)m(or)19 b(suc)o(h)h(a)g(group,)g(the)262 606 y(function)15 b Fn(F)k Ft(=)c Fn(F)546 612 y Fj(G)589 606 y Ft(satis\014es)i Fn(F)777 591 y Fk(0)802 606 y Ft(=)d Fn(F)881 591 y Fl(2)899 606 y Ft(,)i Fn(F)6 b Ft(\(0\))14 b(=)g(1,)h(with)g(solution)g Fn(F)6 b Ft(\()p Fn(t)p Ft(\))14 b(=)h(\(1)10 b Fm(\000)g Fn(t)p Ft(\))1637 591 y Fk(\000)p Fl(1)1682 606 y Ft(.)262 656 y(Th)o(us)15 b Fn(F)395 662 y Fj(n)417 656 y Ft(\()p Fn(G)p Ft(\))f(=)g Fn(n)p Ft(!.)22 b(This)15 b(sequence)i(is)e(the)h(same)f(as)g(the)h (one)f(realised)h(b)o(y)f(the)h(group)262 706 y Fn(A)p Ft(.)h(Indeed,)d(the)g(stabiliser)g(of)f(0)g(in)g Fn(A)g Ft(has)h(t)o(w)o(o)e(orbits,)i(the)f(p)q(ositiv)o(e)h(and)f(the)h (negativ)o(e)262 756 y(rationals;)19 b(eac)o(h)h(orbit,)f(as)g(ordered) h(set,)g(is)f(isomorphic)e(to)h Fi(Q)p Ft(,)f(and)i Fn(A)1460 762 y Fl(0)1497 756 y Ft(induces)h(all)262 805 y(order-preserving)c(p)q (erm)o(utations)f(on)g(eac)o(h.)23 b(So)15 b(indeed)g Fn(G)f Ft(=)g Fn(A)h Ft(satis\014es)i(the)e(original)262 855 y(equation.)i(\(The)12 b(fact)h(that)f Fn(@)r(A)g Ft(=)g Fn(A)6 b Fm(\002)g Fn(A)p Ft(,)12 b(where)i Fn(A)e Ft(is)g(the)h(class)f(of)g(\014nite)g(total)g(orders,)262 905 y(can)j(b)q(e)g(regarded)h(as)f(the)g(basis)g(for)g(the)g(recursiv) o(e)h(QUICKSOR)m(T)f(algorithm)e([15)o(])h(for)262 955 y(sorting)i(a)h(list:)24 b(select)18 b(an)f(elemen)o(t)f(0,)h (partition)f(the)i(list)e(in)o(to)g(elemen)o(ts)h(b)q(efore)h(and)262 1005 y(after)c(0,f(and)g(排序)i(t)o(w)o(o)f(子列表))324(1054)y(e)(f)fn(g)g fn(c)d fn(c)d ft(w)m(r)e fn(s)14 b Ft(也)d(Syss\014es)h fn(f)1009 1060 y fJ(n)1031π1054 y FT(\(p)fn(g)p ft(\))g(=)g fn(n)p fT(n),)e(corresp)q(onding)j (com)o(binator-)262 1104 y(ially)e(to)j(the)g(fact)g(that)g(an)o(y)f(p) q(erm)o(utation)g(can)h(b)q(e)g(decomp)q(osed)g(in)o(to)f(a)g(disjoin)o (t)g(union)262 1154 y(of)g(cycles.)19 b(This)14 b(group,)f(lik)o(e)g Fn(A)h Ft(itself,)f(satis\014es)i(the)g(related)f(equation)g Fn(G)1467 1160 y Fj(\013)1501 1143 y Fm(\030)1501 1156 y Ft(=)1545 1154 y Fn(A)c Fm(\002)f Fn(G)p Ft(.)324 1204 y(What)15 b(ab)q(out)g(the)h(di\013eren)o(tial)f(equation)g Fn(G)1048 1210 y Fj(\013)1085 1204 y Ft(=)f Fn(G)7 b Ft(W)m(r)f Fn(G)p Ft(?)22 b(It)16 b(can)f(b)q(e)h(sho)o(wn)g(that)262 1254 y(no)g(suc)o(h)h(group)g(exists.)27 b(Nev)o(ertheless,)19 b(w)o(e)e(obtain)f(an)g(in)o(teresting)h(in)o(teger)g(sequence)262 1303 y(\()p Fn(F)305 1309 y Fj(n)327 1303 y Ft(\()p Fn(G)p Ft(\)\))d(for)f(suc)o(h)i(a)f(non-existen)o(t)g(group.)k(With)13 b Fn(f)t Ft(\()p Fn(t)p Ft(\))g(=)f Fn(F)1245 1309 y Fj(G)1272 1303 y Ft(\()p Fn(t)p Ft(\))e Fm(\000)f Ft(1,)k(w)o(e)h(ha)o (v)o(e)674 1395 y Fn(f)698 1378 y Fk(0)711 1395 y Ft(\()p Fn(t)p Ft(\))e(=)f(1)e(+)h Fn(f)t Ft(\()p Fn(f)t Ft(\()p Fn(t)p Ft(\)\))p Fn(;)91 b(f)t Ft(\(0\))13 b(=)f(0)p Fn(;)262 1486 y Ft(somewhat)g(reminiscen)o(t)i(of)f(the)i(F)m(eigen)o (baum{Cvitano)o(vi)n(\023)-20 b(c)11 b(equation)801 1577 y Fn(g)q Ft(\()p Fn(t)p Ft(\))h(=)g Fm(\000)p Fn(\013g)q Ft(\()p Fn(g)q Ft(\()p Fn(t=\013)p Ft(\)\))262 1669 y(\(F)m(eigen)o (baum)h([12)o(]\).)21 b(The)16 b(unique)f(p)q(o)o(w)o(er)g(series)i (solution)d(do)q(es)i(not)f(con)o(v)o(erge)g(in)g(an)o(y)262 1719 y(neigh)o(b)q(ourho)q(o)q(d)f(of)g(0.)19 b(Is)c(the)g(a)f(com)o (binatorial)d(in)o(terpretation)k(of)f(the)h(co)q(e\016cien)o(ts)h(\(a) 262 1768 y(class)g(of)f(structures)j(en)o(umerated)e(b)o(y)f(them\)?)23 b(The)16 b(\014rst)h(few)f(terms)f(of)g(the)h(sequence)262 1818 y(are)e(1,)f(2,)g(7,)g(37,)g(269,)g(2535,)f(29738,)g(421790,)g (7076459,)g(.)6 b(.)h(.)20 b(.)262 1955 y Fp(12)66 b(The)22 b(probabilit)n(y)k(of)c(connectedness)262 2046 y Ft(According)d(to)f (Ca)o(yley's)g(Theorem,)h(the)h(n)o(um)o(b)q(er)e(of)g(lab)q(elled)g (trees)j(on)d Fn(n)h Ft(p)q(oin)o(ts)g(is)262 2096 y Fn(n)287 2081 y Fj(n)p Fk(\000)p Fl(2)352 2096 y Ft(.)h(It)14 b(is)h(a)f(surprising)h(fact,)g(pro)o(v)o(ed)f(b)o(y)h(R)o(\023)-20 b(en)o(yi)14 b([27)o(])g(in)g(1959,)f(that)i(the)g(n)o(um)o(b)q(er)f (of)262 2146 y(lab)q(elled)g(forests)h(on)g Fn(n)g Ft(p)q(oin)o(ts)f (is)h(asymptotic)e(to)i Fn(cn)1123 2131 y Fj(n)p Fk(\000)p Fl(2)1188 2146 y Ft(,)f(where)i Fn(c)d Ft(=)1411 2116 y Fm(p)p 1446 2116 19 2 v 30 x Ft(e;)i(that)g(is,)f(the)262 2196 y(probabilit)o(y)d(that)j(a)f(random)f(forest)i(on)g Fm(f)p Ft(1)p Fn(;)7 b Ft(2)p Fn(;)g(:)g(:)g(:)s(;)g(n)p Fm(g)13 b Ft(is)g(connected)j(tends)e(to)f(1)p Fn(=)1590 2166 y Fm(p)p 1625 2166 V 30 x FT(e)h(as)262 2246 y fn(n)jωfM(!)h(1)p Ft(.)28 b(\(I)18 b(am)e(grateful)h(to)h(Dominic)d(W)m(elsh)i (for)h(this)f(reference.\))32 b(Moreo)o(v)o(er,)19 b(for)262 2295 y(lab)q(elled)12 b(forests)i(of)f(ro)q(oted)h(trees,)g(the)g (limiting)c(probabilit)o(y)h(of)i(connectedness)j(is)d(1)p Fn(=)p Ft(e.)324 2345 y(In)18 b(terms)h(of)f(our)h(earlier)g(notation,) f(if)g Fn(C)1020 2351 y Fj(n)1062 2345 y Ft(=)i Fn(n)1139 2330 y Fj(n)p Fk(\000)p Fl(2)1222 2345 y Ft(and)f(\()p Fn(A)1355 2351 y Fj(n)1378 2345 y Ft(\))g(is)f(the)h(sequence)262 2395 y(obtained)d(b)o(y)h(applying)f(the)h(op)q(erator)h(EXP)f(to)g(\() p Fn(C)1118 2401 y Fj(n)1140 2395 y Ft(\),)h(then)f(lim)1341 2401 y Fj(n)p Fk(!1)1437 2395 y Fn(A)1468 2401 y Fj(n)1490 2395 y Fn(=C)1541 2401 y Fj(n)1580 2395 y Ft(=)1629 2365 y Fm(p)p 1664 2365 V 30 x Ft(e.)262 2445 y(And,)c(if)g(w)o(e)h(put)g Fn(C)569 2451 y Fj(n)603 2445 y Ft(=)e Fn(n)672 2430 y Fj(n)p Fk(\000)p Fl(1)751 2445 y Ft(instead,)i(the)g(limit)d(is)j(e.) 957 2574 y(24)p eop %%Page: 25 25 25 24 bop 324 307 a Ft(One)13 b(could)g(ask)f(more)g(generally:)17 b(for)12 b(whic)o(h)h(classes)g(of)f(structures)k(\(with)c(a)g(notion) 262 357 y(of)h(connectedness\))k(is)d(it)g(true)h(that)f(the)h (probabilit)o(y)e(of)g(connectedness)k(for)d(a)g(lab)q(elled)262 407 y(or)c(unlab)q(elled)h(structure)i(tends)f(to)f(a)g(limit)d (strictly)j(b)q(et)o(w)o(een)i(zero)f(and)e(one?)18 b(A)11 b(class)g(of)262 457 y(examples)f(is)h(pro)o(vided)h(b)o(y)f(the)h (N-free)g(graphs.)18 b(As)12 b(w)o(e)f(sa)o(w,)h(exactly)f(half)g(of)g (the)h(N-free)262 506 y(graphs)j(on)h Fn(n)f Ft(p)q(oin)o(ts)g(are)h (connected)i(if)c Fn(n)h(>)f Ft(1,)h(and)h(this)f(is)h(true)g(for)f (lab)q(elled)g(or)h(un-)262 556 y(lab)q(elled)d(structures,)j(since)f (complemen)o(tation)c(giv)o(es)j(a)g(bijection)f(b)q(et)o(w)o(een)j (connected)262 606 y(and)11 b(disconnected)j(structures.)20 b(F)m(urthermore,)12 b(it)g(can)g(b)q(e)h(sho)o(wn)f(that)g(the)h (probabilit)o(y)262 656 y(that)18 b(a)g(\(lab)q(elled)f(or)h(unlab)q (elled\))g(N-free)h(p)q(oset)g(is)f(connected)i(tends)f(to)f(the)h (golden)262 706 y(ratio)13 b(as)h(the)g(n)o(um)o(b)q(er)f(of)h(p)q(oin) o(ts)f(tends)i(to)f(in\014nit)o(y)f(\(see)i([10)o(]\).)324 756 y(In)e(the)h(unlab)q(elled)e(case,)i(it)f(is)g(easy)g(to)g(handle)g (ro)q(oted)h(trees,)g(since)g(the)g(n)o(um)o(b)q(er)e(of)262 805 y(forests)e(of)g(ro)q(oted)g(trees)i(on)e Fn(n)f Ft(v)o(ertices)j(is)e(equal)f(to)h(the)h(n)o(um)o(b)q(er)e(of)g(ro)q (oted)i(trees)g(on)f Fn(n)q Ft(+)q(1)262 855 y(v)o(ertices.)18 b(\(T)m(ak)o(e)11 b(a)h(new)g(ro)q(ot,)f(and)h(join)f(it)g(to)g(all)g (the)h(old)f(ro)q(ots.\))18 b(Since)12 b(these)h(n)o(um)o(b)q(ers)262 905 y(gro)o(w)18 b(exp)q(onen)o(tially)g(with)h(constan)o(t)g(2.95576.) 5 b(.)h(.)25 b([24],)19 b(the)g(limiting)d(probabilit)o(y)h(of)262 955 y(connectedness)h(is)d(the)h(recipro)q(cal)g(of)f(this)h(n)o(um)o (b)q(er,)e(namely)g(0.33832.)5 b(.)h(.)h(.)22 b(It)16 b(app)q(ears)262 1005 y(that)h(exp)q(onen)o(tial)g(gro)o(wth)f(for)h (the)h(n)o(um)o(b)q(er)e(of)h Fn(n)p Ft(-elemen)o(t)f(unlab)q(elled)h (structures)j(is)262 1054 y(necessary)e(for)e(the)h(probabilit)o(y)d (of)i(connectedness)j(to)d(b)q(e)h(strictly)g(b)q(et)o(w)o(een)g(0)f (and)g(1,(O)(E)G(SUC)E(H)(a)(324)1154(y)d(项)g(f)g(群)h(G)(问题)f(b)q(EcMe:)for 18(b)(c)(WHIC)o(H)f(寡形)O(C)E(组)j j fn(g)262 1204 y FT(is)c(it)h(真)G(G)(任一)h(LIM)682πy fj(n)p fk(!)(262)1104 y(())d(i)h(不能)g(Pro)1)778 1204 y fn(f)805 y y fj(n)827 y 1204 y FT(\)(p fn(g)7 bf(w)m(r)g fn(s)r ft(\))p fn(=f)1037 y 1210 fj(n)1060 1204 y ft(\(p)fn(g)p ft(\))j(or)h(LIM)τy fj(n)p fk(!)1)1349 1204 y fn(f)1369 y y fJ(n)1391 y 1204 y FT(\(p)fn(g)c Ft(w)m(r)g fn(s)r ft(\))p fn(=f)1594 y 1210 fy(n)1617 1204 y FT(\(p)fn(g)p ft(\))262 262 y(存在)b b(and)f(is)g(\014nITE)h(and)f(更大)h(大于)g(??)k(Ha)o (ving)13 b(form)o(ulated)g(the)i(question)g(in)f(this)262 1303 y(w)o(a)o(y)m(,)8 b(it)h(immediately)d(generalises.)18 b(W)m(e)9 b(can)h(replace)g(the)h(group)e Fn(S)j Ft(b)o(y)e(an)o(y)f (oligomo)o(rphic)262 1353 y(group,)j(tak)o(e)h(the)h(wreath)f(pro)q (duct)h(in)e(either)i(order,)g(or)f(use)g(direct)h(pro)q(duct)g (instead)f(of)262 1403 y(wreath)h(pro)q(duct.)19 b(F)m(or)13 b(more)g(on)h(this,)f(see)j([10)o(].)262 1540 y Fp(13)66 b(Tw)n(o-graphs)22 b(revisited)262 1631 y Ft(The)11 b(last)f(story)m(,) h(lik)o(e)f(the)i(\014rst,)f(is)g(ab)q(out)g(t)o(w)o(o-graphs,)f(and)h (is)g(tak)o(en)g(from)e(Cameron)g([9],)262 1681 y(whic)o(h)k(con)o (tains)h(all)f(references)j(for)e(this)g(section)h(\(and)e(is)h(a)o(v)n (ailable)e(electronically\).)324 1731 y(There)k(is)e(a)h(simple)e (construction)j(for)f(t)o(w)o(o-graphs)f(from)f(trees,)j(as)f(follo)o (ws.)k(Let)d Fn(T)262 1781 y Ft(b)q(e)d(a)f(tree)i(with)f(edge)g(set)g (\012.)18 b(No)o(w)12 b(let)h Fm(T)23 b Ft(consist)13 b(of)f(all)g(triples)h(of)f(edges)i(whic)o(h)e(do)h(not)262 1831 y(lie)f(on)h(a)g(path)g(in)g(the)h(tree)h(\(those)f(for)f(whic)o (h)g(the)h(paths)f(connecting)h(them)f(in)f(the)i(tree)262 1880 y(form)h(a)h(subtree)j(con)o(taining)d(a)h(triv)n(alen)o(t)f(v)o (ertex\).)28 b(It)17 b(is)g(easily)f(v)o(eri\014ed)i(that)f(\(\012)p Fn(;)7 b Fm(T)j Ft(\))262 1930 y(is)16 b(a)g(t)o(w)o(o-graph)g(\(b)o(y) g(considering)h(the)g(four)f(p)q(ossible)h(con\014gurations)f(of)g (four)g(edges\).)262 1980 y(These)c(t)o(w)o(o-graphs)g(arose)g(in)f (the)h(w)o(ork)f(of)g(Tsarano)o(v)g([37])g(on)g(a)g(class)h(of)f (groups)h(related)262 2030 y(to)h(Co)o(xeter)i(groups.)j(Whic)o(h)c(t)o (w)o(o-graphs)f(are)h(pro)q(duced)i(b)o(y)d(the)i(construction?)324 2080 y(The)j Fo(p)n(entagon)g Ft(and)g Fo(hexagon)h Ft(t)o(w)o (o-graphs)e(refer)i(to)e(the)h(t)o(w)o(o-graphs)f(asso)q(ciated,)262 2129 y(as)e(in)f(the)i(\014rst)g(section,)f(with)g(the)h(switc)o(hing)f (classes)h(of)e(the)i(p)q(en)o(tagon)f(and)g(hexagon)262 2179 y(graphs)f(resp)q(ectiv)o(ely)m(.)21 b(In)14 b([8)o(],)g(I)g(pro)o (v)o(ed)h(that)f(a)h(t)o(w)o(o-graph)e(arises)i(from)e(a)h(tree)i(b)o (y)e(the)262 2229 y(construction)i(describ)q(ed)i(if)d(and)h(only)f(if) g(it)h(do)q(esn't)g(con)o(tain)f(either)i(the)g(p)q(en)o(tagon)f(or)262 2279 y(the)f(hexagon)g(t)o(w)o(o-graph)f(as)h(an)g(induced)g (substructure.)24 b(Moreo)o(v)o(er,)16 b(non-isomorphic)262 2329 y(trees)f(giv)o(e)e(rise)h(to)g(non-isomorphic)e(t)o(w)o (o-graphs.)18 b(This)13 b(solv)o(es)h(the)h(coun)o(ting)e(problem)262 2378 y(for)f(unlab)q(elled)h(p)q(en)o(tagon-)g(and)g(hexagon-free)h(t)o (w)o(o-graphs:)j(the)d(n)o(um)o(b)q(er)f(on)g Fn(n)g Ft(p)q(oin)o(ts)262 2428 y(is)g(equal)h(to)g(the)g(n)o(um)o(b)q(er)f (of)g(trees)j(with)e Fn(n)f Ft(edges,)i(calculated)f(b)o(y)f(Otter)j ([24)o(].)957 2574 y(25)p eop %%Page: 26 26 26 25 bop 324 307 a Ft(Ho)o(w)o(ev)o(er,)16 b(there)i(is)e(a)g(further) h(di\016cult)o(y)e(asso)q(ciated)i(with)f(coun)o(ting)f(the)i(lab)q (elled)262 357 y(p)q(en)o(tagon-)c(and)h(hexagon-free)g(t)o(w)o (o-graphs.)k(F)m(or)13 b(example,)g(a)g(path)h(with)g Fn(n)f Ft(edges)i(can)262 407 y(ha)o(v)o(e)h(its)g(edges)i(lab)q(elled) e(in)g Fn(n)p Ft(!)p Fn(=)p Ft(2)g(di\013eren)o(t)h(w)o(a)o(ys,)g(but)f (all)g(of)g(these)h(giv)o(e)f(rise)i(to)e(the)262 457 y(n)o(ull)c(t)o(w)o(o-graph)i(\(the)g(t)o(w)o(o-graph)f(with)h(no)g (triples\).)324 506 y(The)f(solution)g(to)f(the)i(problem)e(comes)g(b)o (y)h(sho)o(wing)g(that)g(the)h(t)o(w)o(o-graph)e(obtained)262 556 y(from)f(a)i(tree)h Fn(T)19 b Ft(is)13 b(reduced)i(\(in)e(the)g (sense)i(of)d(the)i(\014rst)g(section\))g(if)e(and)h(only)f(if)h(the)g (tree)262 606 y(is)h Fo(series-r)n(e)n(duc)n(e)n(d)p Ft(,)f(that)i(is,)f(has)g(no)h(v)o(ertices)g(of)f(v)n(alency)g(2.)20 b(So)14 b(w)o(e)h(should)f(\014rst)h(coun)o(t)262 656 y(the)f(series-reduced)j(edge-lab)q(elled)d(trees.)20 b(The)15 b(n)o(um)o(b)q(er)e(of)h(these)h(with)f Fn(n)g Ft(edges)h(turns)262 706 y(out)e(to)h(b)q(e)497 826 y Fn(x)521 832 y Fj(n)555 826 y Ft(=)606 798 y(1)p 604 817 25 2 v 604 855 a Fn(n)641 774 y Fj(n)p Fk(\000)p Fl(1)642 787 y Fg(X)643 875 y Fj(j)r Fl(=0)704 826 y Ft(\()p Fm(\000)p Ft(1\))789 809 y Fj(j)807 768 y Fg(\022)837 798 y Fn(n)9 bFT(+)H(1)876 y y fn(j)934 768 y fg(\ 023)o(022)995 y 798 fn(n)f fm(000)g ft(1)1033 y y fn(j)αy y fg(\y)y fn(j)r ft(!)\()p Fn(n)g Ft(+)g(1)g Fm(\000)h Fn(j)r Ft(\))1351 809 y Fj(n)p Fk(\000)p Fl(1)p Fk(\000)p Fj(j)262 959 y Ft(for)g Fn(n)h Fm(\025)h Ft(2,)e(with)g Fn(x)560 965 y Fl(1)590 959 y Ft(=)i(1.)17 b(Then)10 b(the)h(n)o(um)o(b)q(er)f(of)g(lab)q(elled)g(p)q(en)o(tagon-)g(and)g (hexagon-free)262 1009 y(t)o(w)o(o-graphs)j(is)h(giv)o(en)f(b)o(y)h (the)g(STIRLING)f(transform)873 1080 y Fj(n)853 1092 y Fg(X)853 1181 y Fj(k)q Fl(=1)920 1131 y Fn(S)r Ft(\()p Fn(n;)7 b(k)q Ft(\))p Fn(x)1070 1137 y Fj(k)1091 1131 y Fn(:)324 1260 y Ft(W)m(e)k(ha)o(v)o(e)h(a)g(language)f(to)h(describ)q (e)i(this)e(b)q(eha)o(viour.)17 b(W)m(e)12 b(can)g(asso)q(ciate)h(a)e (sequence)262 1310 y(op)q(erator)19 b(with)f(a)h(class)g(of)f(ob)r (jects)i(ev)o(en)g(if)e(it)g(is)h(not)g(the)g(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e)20 b(class)f(asso)q(ciated)262 1360 y(with)14 b(some)f(group:)19 b(de\014ne)d(the)f(\\mo)q(di\014ed)e (cycle)i(index")f(to)g(b)q(e)h(the)g(sum)f(of)g(the)h(cycle)262 1409 y(indices)j(of)e(the)j(automorphism)14 b(groups)k(of)f(the)h (unlab)q(elled)f(structures)j(in)d(the)i(class,)262 1459 y(and)13 b(then)g(use)h(the)g(same)e(formalism)e(as)j(describ)q(ed)i (earlier.)j(No)o(w)13 b(series-reduced)j(trees)262 1509 y(\(coun)o(ted)i(b)o(y)e(edges\))j(and)e(reduced)i(p)q(en)o(tagon-)e (and)g(hexagon-free)g(t)o(w)o(o-graphs)g(ha)o(v)o(e)262 1559 y(the)c(same)e(mo)q(di\014ed)g(cycle)j(index,)e(b)q(ecause)i(of)e (the)h(corresp)q(ondence,)i(and)e(hence)h(de\014ne)262 1609 y(the)h(same)f(sequence)j(op)q(erator.)22 b(If)14 b(w)o(e)i(denote)f(this)g(class)h(b)o(y)e Fn(S)r(RT)6 b Ft(,)16 b(then)f(the)h(class)f(of)262 1659 y(all)f(p)q(en)o(tagon-)h (and)h(hexagon-free)g(t)o(w)o(o-graphs)f(corresp)q(onds)j(to)d Fn(S)10 b Ft(W)m(r)c Fn(S)r(RT)g Ft(,)17 b(and)e(the)262 1708 y(class)i(of)g(all)f(trees)j(to)e Fn(A)7 b Ft(W)m(r)g Fn(S)r(RT)24 b Ft(apart)17 b(from)e(a)i(sligh)o(t)g(mismatc)o(h)d(for)j (paths.)29 b(\(The)262 1758 y(edges)18 b(on)f(a)f(path)i(ha)o(v)o(e)f (t)o(w)o(o)f(p)q(ossible)i(orders)g(whic)o(h)f(cannot)g(b)q(e)h (distinguished,)f(but)262 1808 y(whic)o(h)c(are)i(coun)o(ted)f(t)o (wice)g(b)o(y)g Fn(A)7 b Ft(W)m(r)g Fn(S)r(RT)f Ft(.\))324 1858 y(The)k(class)f(of)g(p)q(en)o(tagon-free)h(t)o(w)o(o-graphs)f (\(those)h(con)o(taining)e(no)h(induced)h(p)q(en)o(tagon\))262 1908 y(is)h(also)f(in)o(teresting.)18 b(It)11 b(is)g(closely)g (connected)i(with)e(the)h(class)f(of)g(N-free)h(graphs;)f(in)g(fact,) 262 1957 y(the)i(op)q(erator)h Fn(@)r Ft(,)f(applied)g(to)f(the)i (class)g(of)e(p)q(en)o(tagon-free)i(t)o(w)o(o-graphs,)f(giv)o(es)f(the) i(class)262 2007 y(of)c(N-free)h(graphs)g(\(lik)o(e)f(the)h(relation)f (b)q(et)o(w)o(een)j(t)o(w)o(o-graphs)d(and)g(graphs\).)18 b(Its)11 b(mem)o(b)q(ers)262 2057 y(can)i(also)h(b)q(e)g(represen)o (ted)i(b)o(y)e(trees)h(\(in)f(a)f(di\013eren)o(t)i(w)o(a)o(y\);)d(and)i (it)f(can)h(b)q(e)h(en)o(umerated)262 2107 y(b)o(y)e(tec)o(hniques)i (similar)d(to)h(those)i(describ)q(ed.)20 b(This)14 b(is)f(also)h(found) f(in)h([8)o(],)f([9)o(].)262 2223 y Fa(End)18 b(note)262 2300 y Ft(Jalaluddin)g(Rumi)f(w)o(as)j(one)f(of)g(the)i(leading)d (Su\014)i(p)q(o)q(ets.)36 b(The)20 b(story)g(of)f(the)h(blind)262 2350 y(p)q(eople)13 b(and)g(the)g(elephan)o(t)g(is)g(common)d(to)j(sev) o(eral)g(other)h(religious)e(traditions,)g(includ-)262 2399 y(ing)h(Quak)o(ers)h(and)g(Buddhists.)957 2574 y(26)p eop %%Page: 27 27 27 26 bop 262 307 a Fp(References)282 398 y Ft([1])20 b(R.)e(A.)i(Bailey)m(,)f(Designs:)30 b(mappings)18 b(b)q(et)o(w)o(een)j (structured)h(sets,)f(pp.)f(22{51)e(in)347 448 y Fo(Surveys)d(in)h (Combinatorics,)e(1989)h Ft(\(ed.)g(J.)f(Siemons\),)f(Cam)o(bridge)f (Univ.)h(Press,)347 498 y(Cam)o(bridge,)f(1989.)282 581 y([2])20 b(R.)13 b(A.)g(Bailey)m(,)g(P)m(.)g(J.)h(Cameron)e(and)i(D.)f (G.)g(F)m(on-Der-Flaass,)g(in)g(preparation.)282 664 y([3])20 b(M.)c(Bernstein)j(and)d(N.)h(J.)g(A.)f(Sloane,)h(Some)e (canonical)i(sequences)i(of)d(in)o(tegers,)347 714 y Fo(Line)n(ar)e(A)o(lgebr)n(a)h(Appl.)p Ft(,)e(to)g(app)q(ear.)282 797 y([4])20 b(R.)11 b(Brauer,)j(On)f(the)g(connection)g(b)q(et)o(w)o (een)h(the)f(ordinary)f(and)h(the)g(mo)q(dular)d(c)o(har-)347 846 y(acters)15 b(of)e(groups)h(of)g(\014nite)g(order,)g Fo(A)o(nn.)h(Math.)f Fh(42)g Ft(\(1941\),)e(926{935.)282 929 y([5])20 b(T.)c(Bridgeman,)g(Lah's)h(triangle)f(|)g(Stirling)g(n)o (um)o(b)q(ers)h(of)f(the)h(third)g(kind,)g(pre-)347 979 y(prin)o(t,)c(July)h(1995.)282 1062 y([6])20 b(P)m(.)11 b(J.)h(Cameron,)e(Cohomological)e(asp)q(ects)14 b(of)d(t)o(w)o (o-graphs,)g Fo(Math.)i(Z.)f Fh(157)g Ft(\(1977\),)347 1112 y(101{119.)282 1195 y([7])20 b(P)m(.)9 b(J.)h(Cameron,)f Fo(Oligomorphic)i(Permutation)g(Gr)n(oups)p Ft(,)g(London)e(Math.)h(So) q(c.)g(Lec-)347 1245 y(ture)15 b(Notes)f Fh(152)p Ft(,)f(Cam)o(bridge)f (Univ)o(ersit)o(y)i(Press,)h(Cam)o(bridge,)d(1990.)282 1328 y([8])20 b(P)m(.)13 b(J.)h(Cameron,)e(Tw)o(o-graphs)h(and)h (trees,)h Fo(Discr)n(ete)g(Math.)f Fh(127)f Ft(\(1994\),)g(63{74.)282 1411 y([9])20 b(P)m(.)15 b(J.)h(Cameron,)f(Coun)o(ting)g(t)o(w)o (o-graphs)h(related)h(to)f(trees,)h Fo(Ele)n(ctr)n(onic)g(J.)f(Com-)347 1461 y(binatorics)d Fh(2)h Ft(\(1995\),)f(#R4.)262 1544 y([10])19 b(P)m(.)13 b(J.)h(Cameron,)e(On)i(the)h(probabilit)o(y)d(of)h (connectedness,)k(in)c(preparation.)262 1627 y([11])19 b(J.)14 b(D.)f(Dixon,)f(p)q(ersonal)i(comm)o(unication)d(\(1985\).)262 1710 y([12])19 b(M.)13 b(J.)g(F)m(eigen)o(baum,)e(Quan)o(titativ)o(e)i (univ)o(ersalit)o(y)g(for)g(a)g(class)h(of)f(nonlinear)g(trans-)347 1760 y(formations,)e Fo(J.)k(Statist.)f(Phys.)g Fh(19)p Ft(,)f(25{52.)262 1843 y([13])19 b(R.)c(F)m(ra)-5 b(\177)-16 b(\020ss)o(\023)c(e,)17 b(Sur)g(certains)g(relations)f(qui)g(g)o(\023) -20 b(en)o(\023)g(eralisen)o(t)17 b(l'ordre)f(des)h(nom)o(bres)f(ra-) 347 1892 y(tionnels,)d Fo(C.)h(R.)h(A)n(c)n(ad.)g(Sci.)g(Paris)e Fh(237)h Ft(\(1953\),)e(540{542.)262 1976 y([14])19 b(D.)13 b(Glynn,)g(Rings)g(of)h(geometries,)f(I,)h Fo(J.)g(Combinatorial)h(The) n(ory)f Ft(\(A\))g Fh(44)g Ft(\(1987\),)347 2025 y(34{48;)e(I)q(I,)h Fo(ibid.)h Ft(\(A\))g Fh(49)g Ft(\(1988\),)e(26{66.)262 2108 y([15])19 b(C.)13 b(A.)h(R.)f(Hoare,)g(Quic)o(ksort,)h Fo(Computer)h(Journal)f Fh(5)f Ft(\(1962\),)g(10{15.)262 2191 y([16])19 b(A.)11 b(Jo)o(y)o(al,)g(Une)h(th)o(\023)-20 b(eorie)13 b(com)o(binatoire)c(des)k(s)o(\023)-20 b(eries)13 b(formelles,)d Fo(A)n(dvanc)n(es)k(Math.)e Fh(42)347 2241 y Ft(\(1981\),)h(1{82.)262 2324 y([17])19 b(I.)d(Lah,)h(Eine)h (neue)g(Art)f(v)o(on)g(Zahlen,)g(ihre)g(Eigensc)o(haften)h(und)f(An)o (w)o(eldung)g(in)347 2374 y(der)d(mathematisc)o(hen)f(Statistik,)g Fo(Mitt.)h(Math.)h(Statistik)e Fh(7)h Ft(\(1955\),)e(203{212.)957 2574 y(27)p eop %%Page: 28 28 28 27 bop 262 307 a Ft([18])19 b(V.)e(A.)h(Lisk)o(o)o(v)o(ec,)g(En)o (umeration)f(of)g(Euler)h(graphs,)h Fo(V)m(esc)l(\026)-17 b(\020)18 b(A)o(kad.)g(Navuk)h(BSSR)347 357 y(Ser.)14 b(F)l(\026)-17 b(\020z{Mat.)15 b(Navuk)g Ft(\(1970\),)d(38{46.)262 440 y([19])19 b(D.)d(Livingstone)g(and)h(A.)f(W)m(agner,)h(T)m (ransitivit)o(y)e(of)i(\014nite)g(p)q(erm)o(utation)e(groups)347 490 y(on)f(unordered)h(sets,)f Fo(Math.)h(Z.)g Fh(90)e Ft(\(1965\),)g(393{403.)262 573 y([20])19 b(H.)g(D.)g(Macpherson,)i (The)f(action)g(of)e(an)i(in\014nite)f(p)q(erm)o(utation)f(group)i(on)f (the)347 623 y(unordered)e(subsets)i(of)c(a)h(set,)h Fo(Pr)n(o)n(c.)g(L)n(ondon)h(Math.)f(So)n(c.)g Ft(\(3\))f Fh(51)g Ft(\(1983\),)g(471{)347 672 y(486.)262 756 y([21])j(C.)9 b(L.)g(Mallo)o(ws)f(and)i(N.)f(J.)g(A.)g(Sloane,)h(Tw)o(o-graphs,)f (switc)o(hing)h(classes,)h(and)e(Euler)347 805 y(graphs)14 b(are)g(equal)g(in)f(n)o(um)o(b)q(er,)g Fo(SIAM)i(J.)g(Appl.)f(Math.)g Fh(28)g Ft(\(1975\),)e(876{880.)262 888 y([22])19 b(T.)12 b(Molien,)554 878 y(\177)549 888 y(Ub)q(er)i(die)f(In)o(v)n(arian)o (ten)f(der)i(lineare)f(Substitutionsgrupp)q(e,)g Fo(Sitzungs-)347 938 y(b)n(er.)h(K\177)-21 b(onig.)15 b(Pr)n(euss.)f(A)o(kad.)h(Wiss.)f Ft(\(1897\),)f(1152{1156.)262 1021 y([23])19 b(J.)9 b(A.)g(Nelder,)i (The)f(analysis)f(of)f(randomized)h(exp)q(erimen)o(ts)h(with)f (orthogonal)f(blo)q(c)o(k)347 1071 y(structure,)14 b Fo(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)g(the)h(R)n(oyal)f(So)n(ciety,)h (Series)f(A)p Ft(,)e Fh(283)h Ft(\(1965\),)f(147{178.)262 1154 y([24])19 b(R.)13 b(Otter,)h(The)h(n)o(um)o(b)q(er)e(of)g(trees,)i Fo(A)o(nn.)g(Math.)f Ft(\(2\))g Fh(49)g Ft(\(1948\),)e(583{599.)262 1237 y([25])19 b(M.)c(P)o(ouzet,)h(Application)e(d'une)h(propri)o(\023) -20 b(et)o(\023)g(e)16 b(com)o(binatoire)e(des)i(parties)f(d'un)g(en-) 347 1287 y(sem)o(ble)e(aux)h(group)q(es)g(et)h(aux)e(relations,)g Fo(Math.)j(Z.)e Fh(150)f Ft(\(1976\),)g(117{134.)262 1370 y([26])19 b(D.)13 b(E.)g(Radford,)f(A)i(natural)f(ring)g(basis)h (for)f(the)h(sh)o(u\017e)g(algebra)f(and)g(an)h(applica-)347 1420 y(tion)f(to)h(group)g(sc)o(hemes,)g Fo(J.)g(A)o(lgebr)n(a)f Fh(58)h Ft(\(1979\),)e(432{454.)262 1503 y([27])19 b(A.)13 b(R)o(\023)-20 b(en)o(yi,)13 b(Some)g(remarks)g(on)h(the)g(theory)h(of) e(trees,)i Fo(Publ.)g(Math.)g(Inst.)g(Hungar.)347 1553 y(A)n(c)n(ad.)g(Sci.)e Fh(4)h Ft(\(1959\),)f(73{85.)262 1636 y([28])19 b(C.)d(Reutenauer,)h Fo(F)m(r)n(e)n(e)g(Lie)g(A)o(lgebr) n(as)p Ft(,)f(London)g(Math.)f(So)q(c.)i(Monographs)f(\(New)347 1685 y(Series\))f Fh(7)p Ft(,)e(Oxford)h(Univ)o(ersit)o(y)g(Press,)h (1993.)262 1768 y([29])k(R.)12 b(W.)g(Robinson,)f(En)o(umeration)h(of)g (Euler)h(graphs,)g Fo(Pr)n(o)n(of)g(T)m(e)n(chniques)i(in)f(Gr)n(aph) 347 1818 y(The)n(ory)g Ft(\(Pro)q(c.)h(Second)g(Ann)f(Arb)q(or)h(Graph) f(Theory)h(Conf.,)e(Ann)h(Arb)q(or)h(1968\),)347 1868 y(147{153,)d(Academic)h(Press,)i(New)f(Y)m(ork)g(1969.)262 1951 y([30])19 b(J.)10 b(J.)g(Seidel,)g(Strongly)g(regular)g(graphs)g (of)g Fn(L)1064 1957 y Fl(2)1083 1951 y Ft(-t)o(yp)q(e)g(and)g(of)g (triangular)f(t)o(yp)q(e,)i Fo(Pr)n(o)n(c.)347 2001 y(Kon,)k(Ne)n (derl.)f(A)o(kad.)h(Wetensch.)f Ft(\(A\))h Fh(70)e Ft(\(1967\),)g (188{196.)262 2084 y([31])19 b(J.)10 b(J.)f(Seidel,)i(A)f(surv)o(ey)h (of)e(t)o(w)o(o-graphs,)h(pp.)g(481{511)e(in)h Fo(Pr)n(o)n(c.)i(Int.)g (Col)r(lo)n(q.)g(T)m(e)n(orie)347 2134 y(Combinatorie)p Ft(,)i(Accad.)h(Naz.)g(Lincei,)f(Roma,)e(1977.)262 2217 y([32])19 b(J.)14 b(J.)h(Seidel,)g(More)g(ab)q(out)g(t)o(w)o(o-graphs,) f(in)g Fo(Combinatorics,)i(Gr)n(aphs)f(and)i(Com-)347 2267 y(plexity)d Ft(\(Pro)q(c.)h(4th)f(Czec)o(h)h(Symp.,)d(Prac)o (hatice)k(1990\),)d(297{308,)f Fo(A)o(nn.)j(Discr)n(ete)347 2316 y(Math.)f Fh(51)f Ft(\(1992\).)262 2399 y([33])19 b(J.)c(J.)h(Seidel)g(and)f(D.)g(E.)g(T)m(a)o(ylor,)g(Tw)o(o-graphs:)21 b(A)16 b(second)g(surv)o(ey)m(,)g(in)g Fo(A)o(lgebr)n(aic)347 2449 y(Metho)n(ds)f(in)g(Gr)n(aph)g(The)n(ory)p Ft(,)e(Szeged,)i(1978.) 957 2574 y(28)p eop %%Page: 29 29 29 28 bop 262 307 a Ft([34])19 b(Idries)i(Shah)f(\(ed.\),)i Fo(World)f(T)m(ales)p Ft(,)g(Harcourt)g(Brace)g(Jo)o(v)n(ano)o(vic)o (h,)g(New)g(Y)m(ork,)347 357 y(1979.)262 440 y([35])e(N.)d(J.)g(A.)g (Sloane,)g Fo(A)h(Handb)n(o)n(ok)h(of)f(Inte)n(ger)g(Se)n(quenc)n(es)p Ft(,)g(Academic)f(Press,)i(New)347 490 y(Y)m(ork,)13 b(1973.)262 573 y([36])19 b(T.)10 b(P)m(.)g(Sp)q(eed)i(and)f(R.)e(A.)i (Bailey)m(,)f(F)m(actorial)f(disp)q(ersion)j(mo)q(dels,)d Fo(Internat.)j(Statist.)347 623 y(R)n(eview)h Fh(55)h Ft(\(1987\),)f(261{277.)262 706 y([37])19 b(S.)11 b(Tsarano)o(v,)g(On)h (a)f(generalization)g(of)g(Co)o(xeter)h(groups,)g Fo(A)o(lgebr)n(a)g (Gr)n(oups)h(Ge)n(om.)347 756 y Fh(6)h Ft(\(1989\),e(281 { 318)957×2574 y(29)p EOP %%TRAILL结尾的用户/结束钩子已知{结束钩子},如果%%EOF