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ID:14357683
大小:1.15 MB
页数:231页
时间:2018-07-28
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1、generatingfunctionologyHerbertS.WilfDepartmentofMathematicsUniversityofPennsylvaniaPhiladelphia,PennsylvaniaCopyright1990and1994byAcademicPress,Inc.Allrightsre-served.ThisInternetEditionmaybereproducedforanyvalideducationalpurposeofaninstitutionofhigherlearning,inwhichc
2、aseonlythereason-ablecostsofreproductionmaybecharged.Reproductionforprotorforanycommercialpurposesisstrictlyprohibited.viPrefaceThisbookisaboutgeneratingfunctionsandsomeoftheirusesindiscretemathematics.ThesubjectissovastthatIhavenotattemptedtogiveacomprehensivediscussi
3、on.InsteadIhavetriedonlytocommunicatesomeofthemainideas.Generatingfunctionsareabridgebetweendiscretemathematics,ontheonehand,andcontinuousanalysis(particularlycomplexvariablethe-ory)ontheother.Itispossibletostudythemsolelyastoolsforsolvingdiscreteproblems.Assuchthereism
4、uchthatispowerfulandmagicalinthewaygeneratingfunctionsgiveuniedmethodsforhandlingsuchprob-lems.Thereaderwhowishedtoomittheanalyticalpartsofthesubjectwouldskipchapter5andportionsoftheearliermaterial.Toomitthosepartsofthesubject,however,islikelisteningtoastereobroadcasto
5、f,say,Beethoven'sNinthSymphony,usingonlytheleftaudiochannel.Thefullbeautyofthesubjectofgeneratingfunctionsemergesonlyfromtuninginonbothchannels:thediscreteandthecontinuous.Seehowtheymakethesolutionofdierenceequationsintochild'splay.Thenseehowthetheoryoffunctionsofacomp
6、lexvariablegives,virtuallybyinspection,theapproximatesizeofthesolution.Theinterplaybetweenthetwochannelsisvitallyimportantfortheappreciationofthemusic.Inrecentyearstherehasbeenavigoroustrendinthedirectionofndingbijectiveproofsofcombinatorialtheorems.Thatis,ifwewanttopr
7、ovethattwosetshavethesamecardinalitythenweshouldbeabletodoitbyexhibitinganexplicitbijectionbetweenthesets.Inmanycasesthefactthatthetwosetshavethesamecardinalitywasdiscoveredintherstplacebygeneratingfunctionarguments.Also,eventhoughbijectiveargumentsmaybeknown,thegenera
8、tingfunctionproofsmaybeshorterormoreelegant.Thebijectiveproofsgiveoneacertainsatisfyingfeelingthatone`re-ally'
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