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1、ASurveyofAlternatingPermutationsRichardP.StanleyAbstract.Apermutationa1a2···anof1,2,...,nisalternatingifa1>a2a4<···.Wesurveysomeaspectsofthetheoryofalternatingpermu-tations,beginningwiththefamousresultofAndr´ethatifEnisthenumberPnofalternatingpermutationsof1,2,...,n,thenExn=secx+
2、tanx.n≥0n!Topicsincluderefinementsandq-analoguesofEn,variousoccurrencesofEninmathematics,longestalternatingsubsequencesofpermutations,umbralenu-merationofspecialclassesofalternatingpermutations,andtheconnectionbetweenalternatingpermutationsandthecd-indexofthesymmetricgroup.Dedicatedto
3、RezaKhosrovshahiontheoccasionofhis70thbirthday1.Basicenumerativeproperties.LetSndenotethesymmetricgroupofallpermutationsof[n]:={1,2,...,n}.Apermutationw=a1a2···an∈Sniscalledalternatingifa1>a2a4<···.Inotherwords,aiai+1foriodd.Similarlywisreversealternatingifa1<
4、a2>a3···.(Someauthorsreversethesedefinitions.)LetEndenotethenumberofalternatingpermutationsinSn.(SetE0=1.)Forinstance,E4=5,correspondingtothepermutations2143,3142,3241,4132,and4231.ThenumberEniscalledanEulernumberbecauseEulerconsideredthenumbersE2n+1,thoughnotwiththecombinatoriald
5、efinitionjustgiven.(Rather,Eulerdefinedthemviaequation(1.3)below.)TheEulernumbersarenottobeconfusedwiththeEuleriannumbers,whichcountpermutationsbynumberofdescents.Theinvolution(1.1)a1a2···an7→n+1−a1,n+1−a2,···,n+1−anonSnshowsthatEnisalsothenumberofreversealternatingpermutationsinSn.Wew
6、riteAltn(respectively,Raltn)forthesetofalternating(respectively,reversealternating)permutationsw∈Sn.ThesubjectofalternatingpermutationsandEulernumbershasbecomesovastthatitisimpossibletogiveacomprehensivesurvey.Wewillconfineourselvestosomehighlightsandtosomespecialtopicsthatwefindespeci
7、allyinteresting.2000MathematicsSubjectClassification.Primary05E10,Secondary05E05.ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunderGrantNo.0604423.Anyopinions,findingsandconclusionsorrecommendationsexpressedinthismate-rialarethoseoftheauthoranddonotnecessarilyreflec
8、tthoseoftheN
