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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.
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ANALYTIC COMBINATORICS
PHILIPPE FLAJOLET
AlgorithmsProject
INRIARocquencourt
78153LeChesnay
France
&
ROBERT SEDGEWICK
DepartmentofComputerScience
PrincetonUniversity
Princeton,NJ08540
USA
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CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi
CambridgeUniversityPress
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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
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thepersistenceoraccuracyofURLsforexternalor
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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
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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
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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
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viii CONTENTS
C.4. Specialdistributions 774
C.5. Convergenceinlaw 776
BIBLIOGRAPHY 779
INDEX 801
Description:Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, bio
