ANALYTIC COMBINATORICS — COMPLEX ASYMPTOTICS (Chapters IV, V, VI, VII) PHILIPPE FLAJOLET & ROBERT SEDGEWICK Algorithms Project Department of Computer Science INRIA Rocquencourt Princeton University 78153 Le Chesnay Princeton, NJ 08540 France USA Zeroth Edition, May 8, 2004 (This temporary version expires on December 31, 2004) ~ ~ ~ ~ ~ ~ i ABSTRACT This booklet develops in about 240 pages the basics of asymptotic enumeration through an approach that revolves around generating functions and complex analysis. Major proper- ties of generating functions that are of interest here are singularities. The text presents the core of the theory with two chapters on complex analytic methods focusing on rational and meromorphic functions as well as two chapters on fundamentals of singularity analysis and combinatorial 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 in graphs, lattice paths, constrained permutations, trees, mappings, walks, and maps. Acknowledgements. This work was supported in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). This booklet would be substantially different (and much less informative) without Neil Sloane’s Encyclopedia of Integer Sequences and Steve Finch’s Mathematical Constants, both available on the internet. Bruno Salvy and Paul Zimmermann have developed algorithms and libraries for combinatorial structures and generating functions that are based 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 text contains Chapters IV, V, VI, and VII; it is a continuation of “Analytic Combinatorics—Symbolic Methods” (by Flajolet & Sedgewick, 2002). Readers are encouraged to check Philippe Flajolet’s web pages for regular updates and new developments. c Philippe Flajolet and Robert Sedgewick 2004. iii PREFACE Analytic Combinatorics aims at predicting precisely the asymptotic properties of struc- tured combinatorial configurations, through an approach that bases itself extensively on analytic methods. Generating functions are the central objects of the theory. Analytic combinatorics starts from an exact enumerative description of combinatorial structures by means of generating functions, which make their first appearance as purely formal algebraic objects. Next, generating functions are interpreted as analytic objects, that is, as mappings of the complex plane into itself. In this context, singularities play a key roleˆ in extracting a function’s coefficients in asymptotic form and extremely precise estimates result for counting sequences. This chain is applicable to a large number of problems of discrete mathematics relative to words, trees, permutations, graphs, and so on. A suitable adaptation of the theory finally opens the way to the analysis of parameters of large random structures. Analytic combinatorics can accordingly be organized based on three components: — Symbolic Methods develops systematic “symbolic” relations between some of the major constructions of discrete mathematics and operations on generating functions 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 are viewed as analytic transformations of the complex domain (as “analytic” also known as“holomorphic” functions). Singularities then appear to be a key deter- minant of asymptotic behaviour. — Random Structures concerns itself with probabilistic properties of large random structures—which properties hold with “high” probability, which laws govern randomness 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 standard enumerative theory. The approach to quantitative problems of discrete mathematics provided by analytic combinatorics 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 “metric” properties of large random structures. Accordingly, the theory is susceptible to many ap- plications, within combinatorics itself, but, perhaps more importantly, within other areas of 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 important motivation 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 iv generating functions. As it turns out, a collection of general (and simple) theorems provide a systematic translation mechanism between generating functions and asymptotic forms of coefficients. Four chapters compose this booklet. Chapter IV serves as an introduction to complex-analytic methods and proceeds with the treatment of meromorphic functions, that is, functions whose only singularities are poles, rational functions being the simplest case. Chapter V develops applications of rational and meromorphic asymptotics of gen- erating functions, with numerous applications related to words and languages, walks and graphs, 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 and has applications to trees and other recursively defined combinatorial classes. Chapter VII, presents applications of singularity analysis to 2-regular graphs and polynomials, trees of various 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.) Contents Chapter IV. Complex Analysis, Rational and Meromorphic Asymptotics 1 IV. 1. Generating functions as analytic objects 2 IV. 2. Analytic functions and meromorphic functions 5 IV. 2.1. Basics 6 IV. 2.2. Integrals and residues. 8 IV. 3. Singularities and exponential growth of coefficients 14 IV. 3.1. Singularities 14 IV. 3.2. The Exponential Growth Formula 18 IV. 3.3. Closure properties and computable bounds 23 IV. 4. Rational and meromorphic functions 28 IV. 4.1. Rational functions 29 IV. 4.2. Meromorphic Functions 31 IV. 5. Localization of singularities 35 IV. 5.1. Multiple singularities 35 IV. 5.2. Localization of zeros and poles 39 IV. 5.3. The example of patterns in words 41 IV. 6. Singularities and functional equations 44 IV. 7. Notes 52 Chapter V. Applications of Rational and Meromorphic Asymptotics 55 V. 1. Regular specification and languages 56 V. 2. Lattice paths and walks on the line. 69 V. 3. The supercritical sequence and its applications 81 V. 4. Functional equations: positive rational systems 87 V. 4.1. Perron-Frobenius theory of nonnegative matrices 88 V. 4.2. Positive rational functions. 90 V. 5. Paths in graphs, automata, and transfer matrices. 94 V. 5.1. Paths in graphs. 94 V. 5.2. Finite automata. 101 V. 5.3. Transfer matrix methods. 107 V. 6. Additional constructions 112 V. 7. Notes 117 Chapter VI. Singularity Analysis of Generating Functions 119 VI. 1. Introduction 120 VI. 2. Coefficient asymptotics for the basic scale 123 VI. 3. Transfers 130 VI. 4. First examples of singularity analysis 134 VI. 5. Inversion and implicitly defined functions 137 VI. 6. Singularity analysis and closure properties 141 v vi CONTENTS VI. 6.1. Functional composition. 141 VI. 6.2. Differentiation and integration 146 VI. 6.3. Polylogarithms 149 VI. 6.4. Hadamard Products 150 VI. 7. Multiple singularities 153 VI. 8. Tauberian theory and Darboux’s method 155 VI. 9. Notes 159 Chapter VII. Applications of Singularity Analysis 161 VII. 1. The “exp–log” schema 162 VII. 2. Simple varieties of trees 168 VII. 2.1. Basic analyses 168 VII. 2.2. Additive functionals 172 VII. 2.3. Enumeration of some non-plane unlabelled trees 176 VII. 2.4. Tree like structures. 179 VII. 3. Positive implicit functions 183 VII. 4. The analysis of algebraic functions 187 VII. 4.1. General algebraic functions 188 VII. 4.2. Positive algebraic systems 198 VII. 5. Combinatorial applications of algebraic functions 203 VII. 5.1. Context-free specifications and languages 204 VII. 5.2. Walks and the kernel method 208 VII. 5.3. Maps and the quadratic method 213 VII. 6. Notes 216 Appendix B. Basic Complex Analysis 219 Bibliography 235 Index 241 CHAPTER IV Complex Analysis, Rational and Meromorphic Asymptotics The shortest path between two truths in the real domain passes through the complex domain. Ð JACQUES HADAMARD 1 Generating functions are a central concept of combinatorial theory. So far, they have been treated as formal objects, that is, as formal power series. The major theme of Chapters I–III