Jeffrey B. Remmel---Generating Functions for Alternating Descents and Alternating major index.pdf

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1、Ann.Comb.16(2012)625–650AnnalsofCombinatoricsDOI10.1007/s00026-012-0150-9PublishedonlineMay6,2012©SpringerBaselAG2012GeneratingFunctionsforAlternatingDescentsandAlternatingMajorIndexJeffreyB.RemmelDepartmentofMathematics,UniversityofCalifornia,SanDiego,LaJolla,CA92093-0112,USAjremmel@ucsd.eduRe

2、ceivedDecember22,2009MathematicsSubjectClassification:05A05,05A15,05E05Abstract.In2008,Chebikinintroducedthealternatingdescentset,AltDes(σ),ofapermu-tationσ=σ1···σninthesymmetricgroupSnasthesetofallisuchthateitheriisoddandσi>σi+1oriisevenandσi<σi+1.Wecanthendefinealtdes(σ)=

3、AltDes(σ)

4、andaltmaj(σ)

5、=∑i∈AltDes(σ)i.Inthispaper,wecomputeageneratingfunctionforthejointdistri-butionofaltdes(σ)andaltmaj(σ)overSn.OurformulaissimilartotheformulaforthejointdistributionofdesandmajoverthesymmetricgroupthatwasfirstprovedbyGessel.WealsocomputesimilargeneratingfunctionsforthegroupsBnandDnandforr-tuplesof

6、permutationsinSn.Finallyweproveageneralextensionoftheseformulasincaseswherewekeeptrackofdescentsonlyatpositionsr,2r,....Keywords:alternatingdescents,alternatingmajorindex,symmetricfunctions1.IntroductionIfσ=σ1···σnisanelementofthesymmetricgroupSnwritteninonelinenotation,thenweletDes(σ)={i:σi>σi

7、+1}andRise(σ)={i:σi<σi+1}.ForanystatementA,wesetχ(A)=1ifAistrueandχ(A)=0ifAisfalse.Thenweshallconsiderthefollowingpermutationstatistics:des(σ)=

8、Des(σ)

9、,ris(σ)=1+

10、Rise(σ)

11、,maj(σ)=∑i,comaj(σ)=n+∑i,i∈Des(σ)i∈Rise(σ)inv(σ)=∑χ(σi>σj),coinv(σ)=∑χ(σi>σj).1≤i

12、tationsinSn,wedefinethecommondescentsetofσ(1),...,σ(s)byComdesσ(1),...,σ(s)=i:σ(j)>σ(j)forj=1,...,sii+1626J.B.Remmelandweletcomdesσ(1),...,σ(s)=

13、Comdesσ(1),...,σ(s)

14、andcommajσ(1),...,σ(s)=i.∑i∈Comdes(σ(1),...,σ(s))ThesetypesofstatisticshavebeenstudiedbyCarlitz,Scoville,andVaughaninthe1

15、970’sandrevisitedbyFedouandRawlingsinthe1990’s[10,11,14,15].Chebikin[12]definedthealternatingdescentsetofapermutationσ=σ1···σnbyAltDes(σ)={2i:σ2i<σ2i+1}∪{2i+1:σ2i+1>σ2i+2}.(1.1)Wethendefinealtdes(σ)=

16、AltDes(σ)

17、,andaltmaj(σ)=∑i.i∈Alt