Analytic Combinatorics PDF
Preview Analytic Combinatorics
“book” — 2008/10/3 — 16:05 — page 1 — #1 ANALYTIC COMBINATORICS Analyticcombinatoricsaimstoenableprecisequantitativepredictionsoftheproper- ties of large combinatorial structures. The theory has emerged over recent decades asessentialbothfortheanalysisofalgorithmsandforthestudyofscientificmodels in many disciplines, including probability theory, statistical physics, computational biology and information theory. With a careful combination of symbolic enumera- tionmethodsandcomplexanalysis,drawingheavilyongeneratingfunctions,results ofsweepinggeneralityemergethatcanbeappliedinparticulartofundamentalstruc- turessuchaspermutations,sequences,strings,walks,paths,trees,graphsandmaps. This account is the definitive treatment of the topic. In order to make it self- contained, the authors give full coverage of the underlying mathematics and give a thoroughtreatmentofbothclassicalandmodernapplicationsofthetheory.Thetextis complementedwithexercises,examples,appendicesandnotesthroughoutthebookto aidunderstanding. Thebookcanbeusedasareferenceforresearchers,asatextbook foranadvancedundergraduateoragraduatecourseonthesubject,orforself-study. PHILIPPE FLAJOLET is Research Director of the Algorithms Project at INRIA Rocquencourt. ROBERT SEDGEWICK is William O. Baker Professor of Computer Science at PrincetonUniversity. “book” — 2008/10/3 — 16:05 — page 2 — #2 “book” — 2008/10/3 — 16:05 — page 3 — #3 ANALYTIC COMBINATORICS PHILIPPE FLAJOLET AlgorithmsProject INRIARocquencourt 78153LeChesnay France & ROBERT SEDGEWICK DepartmentofComputerScience PrincetonUniversity Princeton,NJ08540 USA “book” — 2008/10/3 — 16:05 — page 4 — #4 CAMBRIDGE UNIVERSITY PRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521898065 (cid:2)c P.FlajoletandR.Sedgewick2009 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2009 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-0-521-89806-5hardback CambridgeUniversityPresshasnoresponsibilityfor thepersistenceoraccuracyofURLsforexternalor third-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. “book” — 2008/10/3 — 16:05 — page v — #5 Contents PREFACE ix ANINVITATIONTOANALYTICCOMBINATORICS 1 PartA. SYMBOLICMETHODS 13 I.COMBINATORIALSTRUCTURESANDORDINARYGENERATINGFUNCTIONS 15 I.1. Symbolicenumerationmethods 16 I.2. Admissibleconstructionsandspecifications 24 I.3. Integercompositionsandpartitions 39 I.4. Wordsandregularlanguages 49 I.5. Treestructures 64 I.6. Additionalconstructions 83 I.7. Perspective 92 II.LABELLEDSTRUCTURESANDEXPONENTIALGENERATINGFUNCTIONS 95 II.1. Labelledclasses 96 II.2. Admissiblelabelledconstructions 100 II.3. Surjections,setpartitions,andwords 106 II.4. Alignments,permutations,andrelatedstructures 119 II.5. Labelledtrees,mappings,andgraphs 125 II.6. Additionalconstructions 136 II.7. Perspective 147 III.COMBINATORIALPARAMETERSANDMULTIVARIATEGENERATINGFUNCTIONS 151 III.1. Anintroductiontobivariategeneratingfunctions(BGFs) 152 III.2. Bivariategeneratingfunctionsandprobabilitydistributions 156 III.3. InheritedparametersandordinaryMGFs 163 III.4. InheritedparametersandexponentialMGFs 174 III.5. Recursiveparameters 181 III.6. Completegeneratingfunctionsanddiscretemodels 186 III.7. Additionalconstructions 198 III.8. Extremalparameters 214 III.9. Perspective 218 PartB. COMPLEXASYMPTOTICS 221 IV.COMPLEXANALYSIS,RATIONALANDMEROMORPHICASYMPTOTICS 223 IV.1. Generatingfunctionsasanalyticobjects 225 IV.2. Analyticfunctionsandmeromorphicfunctions 229 v “book” — 2008/10/3 — 16:05 — page vi — #6 vi CONTENTS IV.3. Singularitiesandexponentialgrowthofcoefficients 238 IV.4. Closurepropertiesandcomputablebounds 249 IV.5. Rationalandmeromorphicfunctions 255 IV.6. Localizationofsingularities 263 IV.7. Singularitiesandfunctionalequations 275 IV.8. Perspective 286 V.APPLICATIONSOFRATIONALANDMEROMORPHICASYMPTOTICS 289 V.1. Aroadmaptorationalandmeromorphicasymptotics 290 V.2. Thesupercriticalsequenceschema 293 V.3. Regularspecificationsandlanguages 300 V.4. Nestedsequences,latticepaths,andcontinuedfractions 318 V.5. Pathsingraphsandautomata 336 V.6. Transfermatrixmodels 356 V.7. Perspective 373 VI.SINGULARITYANALYSISOFGENERATINGFUNCTIONS 375 VI.1. Aglimpseofbasicsingularityanalysistheory 376 VI.2. Coefficientasymptoticsforthestandardscale 380 VI.3. Transfers 389 VI.4. Theprocessofsingularityanalysis 392 VI.5. Multiplesingularities 398 VI.6. Intermezzo:functionsamenabletosingularityanalysis 401 VI.7. Inversefunctions 402 VI.8. Polylogarithms 408 VI.9. Functionalcomposition 411 VI.10. Closureproperties 418 VI.11. TauberiantheoryandDarboux’smethod 433 VI.12. Perspective 437 VII.APPLICATIONSOFSINGULARITYANALYSIS 439 VII.1. Aroadmaptosingularityanalysisasymptotics 441 VII.2. Setsandtheexp–logschema 445 VII.3. Simplevarietiesoftreesandinversefunctions 452 VII.4. Tree-likestructuresandimplicitfunctions 467 VII.5. Unlabellednon-planetreesandPo´lyaoperators 475 VII.6. Irreduciblecontext-freestructures 482 VII.7. Thegeneralanalysisofalgebraicfunctions 493 VII.8. Combinatorialapplicationsofalgebraicfunctions 506 VII.9. Ordinarydifferentialequationsandsystems 518 VII.10. Singularityanalysisandprobabilitydistributions 532 VII.11. Perspective 538 VIII.SADDLE-POINTASYMPTOTICS 541 VIII.1. Landscapesofanalyticfunctionsandsaddle-points 543 VIII.2. Saddle-pointbounds 546 VIII.3. Overviewofthesaddle-pointmethod 551 VIII.4. Threecombinatorialexamples 558 VIII.5. Admissibility 564 VIII.6. Integerpartitions 574 “book” — 2008/10/3 — 16:05 — page vii — #7 CONTENTS vii VIII.7. Saddle-pointsandlineardifferentialequations. 581 VIII.8. Largepowers 585 VIII.9. Saddle-pointsandprobabilitydistributions 594 VIII.10. Multiplesaddle-points 600 VIII.11. Perspective 606 PartC. RANDOMSTRUCTURES 609 IX.MULTIVARIATEASYMPTOTICSANDLIMITLAWS 611 IX.1. Limitlawsandcombinatorialstructures 613 IX.2. Discretelimitlaws 620 IX.3. Combinatorialinstancesofdiscretelaws 628 IX.4. Continuouslimitlaws 638 IX.5. Quasi-powersandGaussianlimitlaws 644 IX.6. Perturbationofmeromorphicasymptotics 650 IX.7. Perturbationofsingularityanalysisasymptotics 666 IX.8. Perturbationofsaddle-pointasymptotics 690 IX.9. Locallimitlaws 694 IX.10. Largedeviations 699 IX.11. Non-Gaussiancontinuouslimits 703 IX.12. Multivariatelimitlaws 715 IX.13. Perspective 716 PartD. APPENDICES 719 AppendixA. AUXILIARYELEMENTARYNOTIONS 721 A.1. Arithmeticalfunctions 721 A.2. Asymptoticnotations 722 A.3. Combinatorialprobability 727 A.4. Cycleconstruction 729 A.5. Formalpowerseries 730 A.6. Lagrangeinversion 732 A.7. Regularlanguages 733 A.8. Stirlingnumbers. 735 A.9. Treeconcepts 737 AppendixB. BASICCOMPLEXANALYSIS 739 B.1. Algebraicelimination 739 B.2. Equivalentdefinitionsofanalyticity 741 B.3. Gammafunction 743 B.4. Holonomicfunctions 748 B.5. ImplicitFunctionTheorem 753 B.6. Laplace’smethod 755 B.7. Mellintransforms 762 B.8. Severalcomplexvariables 767 AppendixC. CONCEPTSOFPROBABILITYTHEORY 769 C.1. Probabilityspacesandmeasure 769 C.2. Randomvariables 771 C.3. Transformsofdistributions 772 “book” — 2008/10/3 — 16:05 — page viii — #8 viii CONTENTS C.4. Specialdistributions 774 C.5. Convergenceinlaw 776 BIBLIOGRAPHY 779 INDEX 801
Description:The list of books you might like
