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1、DiscreTeMathematics37(1981)255-26225.2,North-HollandPublishingCompanyFACTORIZATIONOFPERMUTATIONSINTOn-CYCLES*RichardP.STANLEkBqxwtrnentofMathemcrticsMassachusettsInstituteofTechnology,Cambridge,MA02139,USAReceived29March1980Revised24November1980UsingthecharactertheoryofthesymmetricgroupES,,,anexp
2、licitformulaisderivedforthenumberQ(V)ofwaysofwritingapermutationTTES;,,asaproductofkn-cycles.Fromthistheasymptoticexpansionforgk(n)isderived,providedthatwheni.=2,ThasO(logn)fixedpoints.Inparticular,therefollowsaconjectureofWalkupthatif71;,EG,,isanevenpermutationwithnofixedpoint.,thenlim,,,gJn,,)/
3、(n-2)!=2.1.IntroductionLet7rbeanelementofthesymmetricgroupG,ofallpermutationsofann-elementset.Letg&r)bethenumberofk-tuples(ml,...,u,Jotcyclesaioflengthrzsuchthatv=crllluk.Thusgk(v)=0ifeither(a)visanoddpermutationandIIisanoddinteger,or(b)7risodd,niseven,andkiseven,or(c)?riseven,niseven,andkisodd.H
4、usemoller[6,Proposition41attributestoGleasontheresultthatg,(,n)>0foranyevenw.Thefunctiong&rwassubsequentlyconsideredin[l,2,9].Inparticular,Walkup[9,p.3161conjecturedthatlim,,,gz(vJ(n-2)!=2wherem1,w2,...isanysequenceofevenpermutationswithoutfixedpoints,withVcG,,.WewillusethecharactertheoryofG,,tod
5、eriveanexplicitexpressionforgk(v)fromwhichWalkupsconjecturecanbededuced.Moregerlerally,wecanwritedowntheentireasymptoticexpansionofthefunctiongk(n)forfixedk(providedthenumberoffixedpointsof7~remainssmsllwhenk=2).Thetechniqueofcharactertheorywasalsousedin[l,Section31,andsomespecialcasesofourresult
6、soverlapwiththispaper.In[2,Corollary4.81anexplicitexpressionforg2(r)isderived,whichissimplerthanours,andwhichcanalsobeusedtoproveWalkupsconjecture.IamgratefILltotherefereeforcallingmyattentionto[Z].*PartiallysuppoTtedbytheNationalScienceFoundationandBellTeiephoneLaboratories,MurrayHill,NJ.OO12-36
7、5X/81/0000-0000/$02.7501981North-Holland256RXStanley2.CharactertheoryWefirstreviewtheresultsfromcharactertheorythatwewillneed.LetGbeanyfinitegroupand@Gitsgroupalgebraover@.IfCi,1~ist,isaconjugacyclassofG,thenletKi=Cn.c
