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1、Beyondalternatingpermutations:PatternavoidanceinYoungdiagramsandtableauxNihalGowravaram,RaviJagadeesanMentor:JoelBrewsterLewisNovember28,2012AbstractWeinvestigatepatternavoidanceinalternatingpermutationsandgeneralizationsthereof.First,westudypatternavoidanceinanalternatinganalogu
2、eofYoungdia-grams.Inparticular,weextendBabson-West’snotionofshape-WilfequivalencetoapplytoalternatingpermutationsandsogeneralizeresultsofBackelin-West-XinandOuchterlonytoalternatingpermutations.Second,westudypatternavoidanceinthemoregeneralcontextofpermutationswithrestrictedascen
3、tsanddescents.WeconsideraquestionofLewisregardingpermutationsthatarethereadingwordsofthickenedstaircaseYoungtableaux,thatis,permutationsthathavek−1ascentsfollowedbyadescent,followedbyk−1ascents,etcetera.Wedeterminetherelativesizesofthesetsofpattern-avoiding(k−1)-ascentpermutation
4、sintermsoftheforbiddenpattern.Furthermore,inequalitiesinthesizesofsetsofpattern-avoidingpermutationsinthiscontextarisefromfurtherextensionsofshape-equivalencetypeenumerations.1IntroductionThetheoryofpatternavoidanceinpermutationsisconcernedwithenumerativeproblemsandhasconnections
5、tocomputerscience,algebraiccombinatorics,algebraicgeometry,andarXiv:1301.6796v1[math.CO]28Jan2013representationtheory.Thefundamentalquestionistodeterminethenumberofpermutationsofagivenlengththatavoidacertaintypeofforbiddensubsequence.Forexample,theonlypermutationsthatavoid21areth
6、eidentitypermutations.Thetheoryfirstaroseinthestudyofstack-sortablepermutations;forexample,Knuth[6]showedthatstack-sortablepermuta-tionsareexactlythosethatavoidthepattern231.Additionally,generalizedstack-sortablepermutationsarecharacterizedbytheavoidanceoflongerpatterns;foranexpos
7、ition,see[5,Chapter8].MacDonald[10]demonstratedthatvexillarypermutations,objectsofinterestinalgebraiccombinatorics,arecharacterizedby2143-avoidance.Furthermore,LakshmibaiandSandhya[7]provedthatpermutationsthatsimultaneouslyavoid3412and4231indexsmoothSchubertvarieties,whicharestud
8、iedinalgebraicgeometry.BilleyandWarring-
