Algorithmic and Symbolic Combinatorics. An Invitation to Analytic Combinatorics in Several Variables PDF
Preview Algorithmic and Symbolic Combinatorics. An Invitation to Analytic Combinatorics in Several Variables
Texts & Monographs in Symbolic Computation Stephen Melczer Algorithmic and Symbolic Combinatorics An Invitation to Analytic Combinatorics in Several Variables Texts & Monographs in Symbolic Computation A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria Founding Editor Bruno Buchberger, Research Institute for Symbolic Computation, Linz, Austria Series Editor PeterPaule,ResearchInstituteforSymbolicComputation(RISC),JohannesKepler University, Linz, Austria Mathematics is a key technology in modern society. Symbolic Computation is on its way to become a key technology in mathematics. “Texts and Monographs in Symbolic Computation” provides a platform devoted to reflect this evolution. In addition to reporting on developments in the field, the focus of the series also includesapplicationsofcomputeralgebraandsymbolicmethodsinothersubfields ofmathematicsandcomputerscience,and,inparticular,inthenaturalsciences.To provide a flexible frame, the series is open to texts of various kind, ranging from research compendia to textbooks for courses. Indexed by zbMATH. More information about this series at http://www.springer.com/series/3073 Stephen Melczer Algorithmic and Symbolic Combinatorics An Invitation to Analytic Combinatorics in Several Variables 123 StephenMelczer Department ofMathematics University of Pennsylvania Philadelphia, PA,USA ISSN 0943-853X ISSN 2197-8409 (electronic) Texts& Monographsin Symbolic Computation ISBN978-3-030-67079-5 ISBN978-3-030-67080-1 (eBook) https://doi.org/10.1007/978-3-030-67080-1 MathematicsSubjectClassification: 05-XX,68Rxx,68W30 ©SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicatedtothememoriesof PhilippeFlajoletandHerbWilf Foreword Alittleover100yearsago,HardyandRamanujanusedcomplexintegralstoestimate thenumberofpartitionsofalargeinteger.Thisgaveaninklingofadeepconnection between something elementary (counting) and something deep (complex variable theory). The counting problem yields a generating function; coefficients of this generatingfunctioncanbecomputedorapproximatedviaCauchy’sintegralformula. For many decades Hardy and Ramanujan’s work, built on ideas of de Moivre, Bernoulli,Euler,andDirichlet,stoodasalonetourdeforce.Enumerationcontinued toappearelementaryandisolatedfromtherestofmathematics. In the 1950’s Hardy and Ramanujan’s circle method (integrate over a circle approachingthesingularitiesofthegeneratingfunction)wassystematizedsomewhat by Hayman. A few decades later, Flajolet and Odlyzko distilled the process of singularityanalysistoasetofeffectivetransfertheorems,allowingalmostautomatic derivation of asymptotics. Notably, over this span of seventy years, none of the analysesexceededthetechnicaldifficultyofHardyandRamanujan’s. This book introduces a beast of quite a different color: Analytic Combinatorics in Several Variables (ACSV). Generating functions in more than one variable can be quite powerful. They capture joint distributions of combinatorial features, for examplethesizeofthegroundsetofapermutationaswellasitsnumberofcycles, orthelocationandorientationofadominoinarandomtiling,alongwiththesizeof thetiling.TransferringtheideasoftheHardy-Ramanujananalysistothemultivari- atesetting,however,turnsouttosummonmathematicsfromthefourcornersofthe knownmathematicalworld.StartingwithaCauchyintegral,thistimeinseveralvari- ables,oneisledinsimplecasestosaddlepointintegration.Morecomplicatedcases requiresome harmonic analysisand singularitytheory (inverseFourier transforms ofhomogeneousrationalforms).Sometimesthereisanontrivialchoiceofwhereto putthecontour;thisinvokesalgebraictopologyandstratifiedMorsetheory.Making allofthestepseffectiverequirescomputationalalgebraandhomotopymethods. Thestoryisbeautifulbutalsodifficultandeclectic.Melczerfindsawaytotellit understandablyandsimply.Indeed,forMelczer,computationdrivesunderstanding. ToquotefromKnuth,“Scienceiswhatweunderstandwellenoughtoexplaintoa vii viii Foreword computer.”Andso,drawingfromtheauthor’sexpertiseincomputeralgebra,every partofthestoryistoldfromtheviewpointofeffectivecomputation. The book contains deep theorems, yes, but it embodies much more: a tutorial in computer algebra, expertly conceived illustrations, and a very rich collection of examples. Among the examples one finds Kronecker coefficients, rational period integrals, and models from statistical physics. The author’s background in lattice walk enumeration keeps the book grounded in yet another source of compelling examples.Asthetitleimplies,thisbooksucceedswhereourownarguablyhasnot, in making this material inviting. Those taking up the invitation to ACSV will be expertlyguidedintobeautifulnewterritory. February2020, RobinPemantleandMarkWilson Preface Ifyouaskamathematicianwhattheylovemostaboutmathematics,certainanswers invariably arise: beauty, abstraction, creativity, logical structure, connection (be- tweendisciplinesandbetweenpeople),elegance,applicability,andfun.Thisbook can be viewed as a humble attempt to show that combinatorics is the branch of mathematics best situated to embody and illustrate all of these virtues. Our core subject is the large-scale behaviour of combinatorial objects, with a focus on two goals:thecalculationofapproximateasymptoticbehaviourofsequencesarisingin combinatorialcontexts,andthederivationoflimittheoremsdescribinghowthepa- rametersofcombinatorialobjectsbehaveforrandomobjectsoflargesize.Webegin withasimpleidea,thattostudyasequenceweshouldlookatitsgeneratingfunction (theformalserieswhosecoefficientsarethesequenceofinterest).Algebraic,differ- ential, and functional equations satisfied by the generating function then represent datastructureswhichencodethesequence,withmorecomplicatedsequencesrequir- ingmorecomplicatedencodings.Theseencodingscanbeusedtocomputationally manipulatethesequenceand,inmanycases,determineitsunderlyingproperties. Thefieldofanalyticcombinatoricsstudiestheasymptoticbehaviourofunivari- ate sequences by applying techniques from complex analysis to their generating functions.Theuniversalityofmanypropertiesofanalyticfunctionsoftenallowsfor an automatic asymptotic computation, with much of the theory now implemented in computer algebra systems. In this sense the study of analytic combinatorics for univariate sequences is somewhat classical; it is captured in glorious detail by the bookAnalyticCombinatoricsofPhilippeFlajoletandRobertSedgewick.Morere- cently, in the early twenty-first century, Robin Pemantle and Mark Wilson (later joinedbyYuliyBaryshnikov)combinedmethodsfromcomplexanalysis,singularity theory,algebraicanddifferentialgeometry,andtopologytoformatheoryofanalytic combinatoricsinseveralvariables(ACSV).ThetextbookAnalyticCombinatoricsin SeveralVariables,byPemantleandWilson,providesacomprehensiveoverviewof thesubject,butitsuseofadvancedconstructionscomingfromthevariousmathemat- icaldisciplinesconscriptedintotheirworkmakesitsuitablemainlyforresearchers withstrongmathematicalbackgroundsacrossseveraldomains. ix x Preface The aim of this book is to provide a more accessible introduction to this vast and beautiful area of combinatorics. There are several (potentially overlapping) audiences: mathematicians interested in the behaviour of functions satisfying cer- tain algebraic, differential, or functional equations; combinatorialists interested in learningthetheoryofanalyticcombinatoricsandanalyticcombinatoricsinseveral variables;computerscientistsinterestedinthecomputationalaspectsofthesesub- jects; and researchers from a variety of domains with an interest in the resulting applications. Our presentation is calibrated for an audience of math and computer science graduate students and researchers, although advanced undergraduates and those from adjacent research areas shouldn’t be scared away. Previous knowledge ofsequencesandseries,atthelevelofanadvancedcalculuscourseorfirstcourse in analysis, is the main prerequisite. Of all the advanced mathematical topics en- countered, a familiarity and comfort with the basics of complex analysis (analytic functions, residues, the Cauchy integral formula) are the most crucial, although thenecessarycomplexanalyticbackgroundisreviewedinanappendix.Additional knowledgefromcommutativealgebra,singularitytheory,algebraicanddifferential geometry,andtopologycanhelpputcertainresultsincontextbutarenotassumed. Applications from numerous mathematical and scientific domains are given, in- cluding many illustrations of the various techniques on lattice path enumeration problems.Weputastrongfocusoncomputation,andthecompanionwebsitetothis textcontainscomputeralgebracodeworkingthroughtheexamplescontainedhere. Becauseitdrawsfromsomanydifferentareasofmathematics,analyticcombina- toricsinseveralvariableshasareputationofbeingpowerfulyetimpenetrable.My deepestwishisthatthisworkilluminatesthevastrichesofACSVandopensupthe areatonewresearchersacrossdisciplines. Acknowledgments First and foremost, I owe a great debt to the three architects of analytic combinatorics in several variables, Robin Pemantle, Mark Wilson, and Yuliy Baryshnikov, for their support and guidance over the last several years. This book grew out of my doctoral thesis for the University of Waterloo and the École normalesupérieuredeLyon,whichwasmadeincalculablybetterbythesupervision ofBrunoSalvyandGeorgeLabahn,andbythethesisreportersandmembersofthe thesiscommittee:JasonBell,SylvieCorteel,MichaelDrmota,IraGessel,andÉric Schost. Early versions of this manuscript were used as the basis for short lecture seriesattheUniversityofIllinoisUrbana-ChampaignandtheResearchInstitutefor Symbolic Computation at JKU Linz, and for graduate courses at the University of PennsylvaniaandtheUniversitéduQuébecàMontréal,andIthankallthestudents fromthoseclassesfortheirfeedback.RussellMaygaveinvaluablecommentsonthe text,helpingtoimproveitspresentationandfixmanytypographicalerrors.Marcus Michelen, Shaoshi Chen, Chaochao Zhu, Manuel Kauers, Alin Bostan, and Marc Mezzarobba gave feedback on various versions of the manuscript, along with my summerstudentsKeithRitchieandAndrewMartin.IwouldalsoliketothankPersi Diaconis,JessicaKhera,ErikLundberg,andArminStraubforsuggestingproblems which appear here. I originally became interested in analytic combinatorics as an undergraduatestudent,underprojects(someofwhichinspiredproblemsinthistext) supervisedbyMarniMishna,AlinBostan,andManuelKauers,andIhopethistext
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