Williams---A conjecture of Stanley on alternating permutations.pdf

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ASAbstra t.Wegivetwosimpleproofsofa onje tureofRi hardStanley on erningtheequidistributionofderangementsandalternatingpermutationswiththemaximalnumberofxedpoints.1.
ntrodu tionWewrite[n]={1,...,n}andSn

2、forthesetofpermutationsof[n].Apermu-tationisalternatingifa1>a2a4<....Similarly,denewtobereversealternatingifa1a3....
n[2℄,Ri hardStanleyusedthetheoryofsymmetri fun tionstoenumeratevarious lassesofalternatingpermutationswof{1,2,...n}.ne lassthathe onsideredwereal

3、ternatingpermutationswwithaspe iednumberofxedpoints.∗Writedk(n)(respe tively,dk(n))forthenumberofalternating(respe tively,reversealternating)permutationsinSnwithkxedpoints.Asobservedin[2℄,itisnothardtoseethatmax{k:dk(n)6=0}=⌈n/2⌉,n≥4max{k:d∗(n)6=0}=⌈(n+1)/2⌉,n≥5.kStanley o

4、nje turedthefollowingin[2℄.Conje ture1.[2,Conje ture6.3℄etDndenotethenumberofderangements(permutationswithoutxedpoints)inSn.Thend⌈n/2⌉(n)=D⌊n/2⌋,n≥4∗d⌈(n+1)/2⌉(n)=D⌊(n−1)/2⌋,n≥5.arXiv:math/0702808v3[math.CO]18May2007
nthisnotewewillgivetwoproofsofhis onje turerelatingthenum

5、berofderangementstothenumberofalternatingpermutationswiththemaximalnumberofxedpoints.Bothproofsusethesamebije tionΨ.Therstproofworksdire tlywithpermutationsandshowsthatΨisinje tiveandsurje tive.These ondproofworkswithpermutationtableaux, ertaintableauxwhi harenaturallyinbij

6、e tionwithpermutations,andexpli itly onstru tstheinversetoΨ.Thebije tion(foralternatingpermutations)isillustratedinFigure1,intermsofbothpermutationsandpermutationtableaux.Date:February2,2008.eywordsandphrases.alternatingpermutations,derangements,e-tableau,permutationtableau

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AS314243212143423186755231867482645371431234212413826451736284537132845176412334122341823154766284517332548671Figure1.Thebije tionΨforn=82.ThefirstproofAswewillshowsubsequently,themain asethatoneneedsto onsider on ernsalternatingpermutationsona

8、nevennumberofletters.Theorem2.Forea hnonnegativeintegermdm(2m)=Dm.r