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1、ENUMERATIONOFSNAKESANDCYCLE-ALTERNATINGPERMUTATIONSMATTHIEUJOSUAT-VERGES`TothememoryofVladimirArnol’dAbstract.SpringernumbersareananalogofEulernumbersforthegroupofsignedpermutations.Arnol’dshowedthattheycountsomeobjectscalledsnakes,thatgeneralizealternatingpermutations.Hoffmanestablishedalinkbetw
2、eenSpringernumbers,snakes,andsomepolynomialsrelatedwiththesuccessivederivativesoftrigonometricfunctions.Thegoalofthisarticleistogivefurthercombinatorialpropertiesofderivativepolynomials,intermsofsnakesandotherobjects:cycle-alternatingpermutations,weightedDyckorMotzkinpaths,increasingtreesandfore
3、sts.Weobtainthegen-eratingfunctions,intermsoftrigonometricfunctionsforexponentialonesandintermsofJ-fractionsforordinaryones.Wealsodefinenaturalq-analogs,makealinkwithnormalorderingproblemsandcombinatorialtheoryofdifferentialequations.1.IntroductionItiswell-knownthattheEulernumbersEndefinedbyX∞nzEn=
4、tanz+secz(1)n!n=0countalternatingpermutationsinSn,i.e.σsuchthatσ1>σ2<σ3>...σn.Thestudyofthese,aswellasotherclassesofpermutationscountedbyEn,isavasttopicinenumerativecombinatorics,seethesurveyofStanley[21].ByaconstructionduetoSpringer[20],thereisanintegerK(W)definedforanyCoxetergroupWsuchthatK(Sn)
5、=En.Asforthegroupsofsignedpermutations,Arnol’d[1]introducedsomeparticularkindofsignedpermutationscalledsnakes,countedbythenumbersK(SBn)andK(SDn).Thussnakescanbeconsideredasa“signedanalog”ofalternatingpermutations.Algorithmically,Arnol’d[1]givesamethodtocomputetheseintegersarXiv:1011.0929v1[math.
6、CO]3Nov2010withrecurrencesorganizedintriangulararrayssimilartotheSeidel-EntrigertriangleofEulernumbersEn(seeforexample[11]).Inthisarticle,wearemostlyinterestedinthenumbersSn=K(SBn),whichwerepresviouslyconsideredbyGlaisher[9,§§109and119]inanothercontext.Springer[20]showsthattheysatisfyX∞nz1Sn=,(2
7、)n!cosz−sinzn=0andwecallSnthenthSpringernumber(althoughthereisintheoryaSpringernumberassociatedwitheachCoxetergroup,thisnamewithoutfurtherspecificationusuallyDate:November4,2010.2000MathematicsSubjectClassification.Primary:05A
