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1、Aq-ENUMERATIONOFALTERNATINGPERMUTATIONSMATTHIEUJOSUAT-VERGES`Abstract.AclassicalresultofEulerstatesthatthetangentnumbersareanalternatingsumofEuleriannumbers.AdualresultofRosellestatesthatthesecantnumberscanbeobtainedbyasignedenumerationofderangements.Weshowthatbothidentitiescanberefined
2、withthefollowingstatistics:thenumberofcrossingsinpermutationsandderangements,andthenumberofpatterns31-2inalternatingpermutations.UsingpreviousresultsofCorteel,Rubey,Prellberg,andtheauthor,wederiveclosedformulasforbothq-tangentandq-secantnumbers.Therearetwodifferentmethodstoobtainthesefo
3、rmulas:onewithpermutationtableauxandonewithweightedMotzkinpaths(Laguerrehistories).1.IntroductionTheclassicalEulernumbersEnaregivenbytheTaylorexpansionofthetangentandsecantfunctions:X∞xntan(x)+sec(x)=En.n!n=0Sincethetangentisanoddfunctionandthesecantisanevenfunction,theintegersE2n+1are
4、calledthetangentnumbersandtheintegersE2narecalledthesecantnumbers.Morethanacenturyago,D´esir´eAndr´e[1]showedthatEnisthenumberofalternatingpermutationsinSn.Severalq-analogsofthesenumbershavebeenstudied,mainlybyAndrews,Foata,Gessel,Han[2,3,11,15].Theonesweconsiderhereappearedinanarticle
5、ofHan,Randrianarivony,Zeng[15].Definition1.Theq-tangentnumbersE2n+1(q)aredefinedby:X∞1n(1)E2n+1(q)x=,[1]q[2]qxn=01−[2]q[3]qx1−[3]q[4]qxarXiv:0907.1004v1[math.CO]6Jul20091−...n1−qwhere[n]q=1−q.Andtheq-secantnumbersE2n(q)aredefinedby:X∞1(2)E(q)xn=.2n2[1]qxn=01−[2]2xq1−[3]2xq1−...Date:July6,
6、2009.PartiallysupportedbytheANRJeuneChercheurIComb.AshortversionofthisworkwaspresentedattheconferencePermutationPatterns2009.12MATTHIEUJOSUAT-VERGES`Thesecontinuedfractionsarethenaturalq-analogsoftheonesappearinginFla-jolet’scelebratedarticle[10].Themethodsinthisreferencegivesacombinat
7、orialinterpretationofEn(q)intermsofweightedDyckpaths.Indeed,ifδ∈{0,1}thenE2n+δ(q)isthesumofweightsofDyckpathsoflength2nsuchthat:•theweightofastepրstartingatheighthisqiforsomei∈{0,...,h},•theweightofastepցstartingatheighthisqiforsomei∈{0,...,h−1+δ},andtheweightofthepathistheproductoft
