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Algorithms Project Department of Computer ScienceINRIA Rocquencourt Princeton University78153 Le Chesnay Princeton, NJ 08540France USA

Zeroth Edition, May 8, 2004(This temporary version expires on December 31, 2004)



This booklet develops in about 240 pages the basics of asymptotic enumeration through anapproach that revolves around generating functions and complex analysis. Major proper-ties of generating functions that are of interest here are singularities. The text presents thecore of the theory with two chapters on complex analytic methods focusing on rational andmeromorphic functions as well as two chapters on fundamentals of singularity analysis andcombinatorial consequences. It is largely oriented towards applications of complex anal-ysis to asymptotic enumeration and asymptotic properties of random discrete structures.Many examples are given that relate to words, integer compositions, paths and walks ingraphs, lattice paths, constrained permutations, trees, mappings, walks, and maps.

Acknowledgements. This work was supported in part by the IST Programme of the EU undercontract number IST-1999-14186 (ALCOM-FT). This booklet would be substantially different (andmuch less informative) without Neil Sloanes Encyclopedia of Integer Sequences and Steve FinchsMathematical Constants, both available on the internet. Bruno Salvy and Paul Zimmermann havedeveloped algorithms and libraries for combinatorial structures and generating functions that arebased on the MAPLE system for symbolic computations and have proven to be immensely useful.

This is a set of lecture notes that are a component of a wider book project titled Analytic Combi-natorics, which will provide a unified treatment of analytic methods in combinatorics. This textcontains Chapters IV, V, VI, and VII; it is a continuation of Analytic CombinatoricsSymbolicMethods (by Flajolet & Sedgewick, 2002). Readers are encouraged to check Philippe Flajoletsweb pages for regular updates and new developments.

cPhilippe Flajolet and Robert Sedgewick 2004.



Analytic Combinatorics aims at predicting precisely the asymptotic properties of struc-tured combinatorial configurations, through an approach that bases itself extensively onanalytic methods. Generating functions are the central objects of the theory.

Analytic combinatorics starts from an exact enumerative description of combinatorialstructures by means of generating functions, which make their first appearance as purelyformal algebraic objects. Next, generating functions are interpreted as analytic objects, thatis, as mappings of the complex plane into itself. In this context, singularities play a key rolein extracting a functions coefficients in asymptotic form and extremely precise estimatesresult for counting sequences. This chain is applicable to a large number of problems ofdiscrete mathematics relative to words, trees, permutations, graphs, and so on. A suitableadaptation of the theory finally opens the way to the analysis of parameters of large randomstructures.

Analytic combinatorics can accordingly be organized based on three components:

Symbolic Methods develops systematic symbolic relations between some ofthe major constructions of discrete mathematics and operations on generatingfunctions which exactly encode counting sequences.

Complex Asymptotics elaborates a collection of methods by which one can ex-tract asymptotic counting informations from generating functions, once these areviewed as analytic transformations of the complex domain (as analytic alsoknown asholomorphic functions). Singularities then appear to be a key deter-minant of asymptotic behaviour.

Random Structures concerns itself with probabilistic properties of large randomstructureswhich properties hold with high probability, which laws governrandomness in large objects? In the context of analytic combinatorics, this cor-responds to a deformation (adding auxiliary variables) and a perturbation (exam-ining the effect of small variations of such auxiliary variables) of the standardenumerative theory.

The approach to quantitative problems of discrete mathematics provided by analyticcombinatorics can be viewed as an operational calculus for combinatorics. The booklets,of which this is the second installment, expose this view by means of a very large num-ber of examples concerning classical combinatorial structures (like words, trees, permuta-tions, and graphs). What is aimed at eventually is an effective way of quantifying metricproperties of large random structures. Accordingly, the theory is susceptible to many ap-plications, within combinatorics itself, but, perhaps more importantly, within other areasof science where discrete probabilistic models recurrently surface, like statistical physics,computational biology, or electrical engineering. Last but not least, the analysis of algo-rithms and data structures in computer science has served and still serves as an importantmotivation in the development of the theory.

The present booklet specifically exposes Complex Asymptotics, which is a unified an-alytic theory dedicated to the process of extractic asymptotic information from counting


generating functions. As it turns out, a collection of general (and simple) theorems providea systematic translation mechanism between generating functions and asymptotic formsof coefficients. Four chapters compose this booklet. Chapter IV serves as an introductionto complex-analytic methods and proceeds with the treatment of meromorphic functions,that is, functions whose only singularities are poles, rational functions being the simplestcase. Chapter V develops applications of rational and meromorphic asymptotics of gen-erating functions, with numerous applications related to words and languages, walks andgraphs, as well as permutations. Chapter VI develops a general theory of singularity anal-ysis that applies to a wide variety of singularity types like square-root or logarithmic andhas applications to trees and other recursively defined combinatorial classes. Chapter VII,presents applications of singularity analysis to 2-regular graphs and polynomials, trees ofvarious sorts, mappings, context-free languages, walks, and maps. It contains in particu-lar a discussion of the analysis of coefficients of algebraic functions. (A future chapter,Chapter VIII, will explore saddle point methods.)


Chapter IV. Complex Analysis, Rational and Meromorphic Asymptotics 1IV. 1. Generating functions as analytic objects 2IV. 2. Analytic functions and meromorphic functions 5IV. 2.1. Basics 6IV. 2.2. Integrals and residues. 8IV. 3. Singularities and exponential growth of coefficients 14IV. 3.1. Singularities 14IV. 3.2. The Exponential Growth Formula 18IV. 3.3. Closure properties and computable bounds 23IV. 4. Rational and meromorphic functions 28IV. 4.1. Rational functions 29IV. 4.2. Meromorphic Functions 31IV. 5. Localization of singularities 35IV. 5.1. Multiple singularities 35IV. 5.2. Localization of zeros and poles 39IV. 5.3. The example of patterns in words 41IV. 6. Singularities and functional equations 44IV. 7. Notes 52

Chapter V. Applications of Rational and Meromorphic Asymptotics 55V. 1. Regular specification and languages 56V. 2. Lattice paths and walks on the line. 69V. 3. The supercritical sequence and its applications 81V. 4. Functional equations: positive rational systems 87V. 4.1. Perron-Frobenius theory of nonnegative matrices 88V. 4.2. Positive rational functions. 90V. 5. Paths in graphs, automata, and transfer matrices. 94V. 5.1. Paths in graphs. 94V. 5.2. Finite automata. 101V. 5.3. Transfer matrix methods. 107V. 6. Additional constructions 112V. 7. Notes 117

Chapter VI. Singularity Analysis of Generating Functions 119VI. 1. Introduction 120VI. 2. Coefficient asymptotics for the basic scale 123VI. 3. Transfers 130VI. 4. First examples of singularity analysis 134VI. 5. Inversion and implicitly defined functions 137VI. 6. Singularity analysis and closure properties 141



VI. 6.1. Functional composition. 141VI. 6.2. Differentiation and integration 146VI. 6.3. Polylogarithms 149VI. 6.4. Hadamard Products 150VI. 7. Multiple singularities 153VI. 8. Tauberian theory and Darbouxs method 155VI. 9. Notes 159

Chapter VII. Applications of Singularity Analysis 161VII. 1. The explog schema 162VII. 2. Simple varieties of trees 168VII. 2.1. Basic analyses 168VII. 2.2. Additive functionals 172VII. 2.3. Enumeration of some non-plane unlabelled trees 176VII. 2.4. Tree like structures. 179VII. 3. Positive implicit functions 183VII. 4. The analysis of algebraic functions 187VII. 4.1. General algebraic functions 188VII. 4.2. Positive algebraic systems 198VII. 5. Combinatorial applications of algebraic functions 203VII. 5.1. Context-free specifications and languages 204VII. 5.2. Walks and the kernel method 208VII. 5.3. Maps and the quadratic method 213VII. 6. Notes 216

Appendix B. Basic Complex Analysis 219

Bibliography 235

Index 241


Complex Analysis, Rational andMeromorphic Asymptotics

The shortest path between two truths in the real domainpasses through the complex domain.


Generating functions are a central concept of combinatorial theory. So far, they havebeen treated as formal objects, that is, as formal power series. The major theme of ChaptersIIII has indeed been to demonstrate how the algebraic structure of generating functionsdirectly reflects the structure of combinatorial classes. From now on, we examine gener-ating functions in the light of analysis. This means assigning value

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