Quadrant marked mesh patterns in alternating permutations II.pdf

《Quadrant marked mesh patterns in alternating permutations II.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、QuadrantmarkedmeshpatternsinalternatingpermutationsIISergeyKitaevJeffreyRemmelUniversityofStrathclydeDepartmentofMathematicsLivingstoneTower,26RichmondStreetUniversityofCalifornia,SanDiegoGlasgowG11XH,UnitedKingdomLaJolla,CA92093-0112.USAsergey.kitaev@cis.strath.ac.ukjremmel@ucsd.eduSubmitted:

2、Date1;Accepted:Date2;Published:Date3.MRSubjectClassifications:05A15,05E05AbstractThispaperiscontinuationofthesystematicstudyofdistributionofquadrantmarkedmeshpatternsinitiatedin[7].Westudyquadrantmarkedmeshpatternsonup-downanddown-uppermutations.Keywords:permutationstatistics,markedmeshpattern

3、,distribution1IntroductionThenotionofmeshpatternswasintroducedbyBr¨and´enandClaesson[4]toprovideexplicitexpansionsforcertainpermutationstatisticsas,possiblyinfinite,linearcombinationsof(classical)permutationpatterns(see[6]foracomprehensiveintroductiontothetheoryofpermutationpatterns).Thisnotio

4、nwasfurtherstudiedin[3,5,7,8,9,10,13].arXiv:1207.2008v1[math.CO]9Jul2012Letσ=σ1...σnbeapermutationinthesymmetricgroupSnwritteninone-linenotation.Thenwewillconsiderthegraphofσ,G(σ),tobethesetofpoints(i,σi)fori=1,...,n.Forexample,thegraphofthepermutationσ=471569283ispicturedinFigure1.Thenifwedr

5、awacoordinatesystemcenteredatapoint(i,σi),wewillbeinterestedinthepointsthatlieinthefourquadrantsI,II,III,andIVofthatcoordinatesystemaspicturedinFigure1.Foranya,b,c,d∈NwhereN={0,1,2,...}isthesetofnaturalnumbersandanyσ=σ1...σn∈Sn,wesaythatσimatchesthequadrantmarkedmeshpatternMMP(a,b,c,d)inσifin

6、G(σ)relativetothecoordinatesystemwhichhasthepoint(i,σi)asitsorigin,thereare≥apointsinquadrantI,≥bpointsinquadrantII,≥cpointsinquadrantIII,and≥dpointsinquadrantIV.Forexample,ifσ=471569283,thepointσ4=5matchesthequadrantmarkedmeshpatternMMP(2,1,2,1)sincerelativetothe1coordinatesystemwithorigin(4

7、,5),thereare3pointsinG(σ)inquadrantI,1pointinG(σ)inquadrantII,2pointsinG(σ)inquadrantIII,and2pointsinG(σ)inquadrantIV.NotethatifacoordinateinMMP(a,b,c,d)is0,thenthereisnoconditionimposedonthepointsinthecorrespondingquadrant.Inaddition,weshall