Lectures on Generating Functions

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This book, based on the course given by the author at the College of Mathematics of the Independent University of Moscow, introduces the reader to the language of generating functions, which is nowadays the main language of enumerative combinatorics. It starts with definitions, simple properties, and numerous examples of generating functions. It then discusses topics, such as formal grammars, generating functions in several variables, partitions and decompositions, and theexclusion-inclusion principle. In the final chapter, the author describes applications of generating functions to enumeration of trees, plane graphs, and graphs embedded in two-dimensional surfaces. Throughout the book, the reader is motivated by interesting examples rather than by general theories. It alsocontains a lot of exercises to help the reader master the material. Little beyond the standard calculus course is necessary to understand the book. It can serve as a text for a one-semester undergraduate course in combinatorics.

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Preface to the English Edition	p. xi
Preface	p. xiii
Formal Power Series and Generating Functions. Operations with Formal Power Series. Elementary Generating Functions	p. 1
The lucky tickets problem	p. 1
First conclusions	p. 6
Generating functions and operations with them	p. 7
Elementary generating functions	p. 10
Differentiating and integrating generating functions	p. 12
The algebra and the topology of formal power series	p. 13
Problems	p. 14
Generating Functions for Well-known Sequences	p. 17
Geometric series	p. 17
The Fibonacci sequence	p. 18
Recurrence relations and rational generating functions	p. 21
The Hadamard product of generating functions	p. 23
Catalan numbers	p. 25
Problems	p. 30
Unambiguous Formal Grammars. The Lagrange Theorem	p. 35
The Dyck Language	p. 35
Productions in the Dyck language	p. 36
Unambiguous formal grammars	p. 38
The Lagrange equation and the Lagrange theorem	p. 42
Problems	p. 43
Analytic Properties of Functions Represented as Power Series and the Asymptotics of their Coefficients	p. 47
Exponential estimates for asymptotics	p. 47
Asymptotics of hypergeometric sequences	p. 50
Asymptotics of coefficients of functions related by the Lagrange equation	p. 54
Asymptotics of coefficients of generating series and singularities on the boundary of the disc of convergence	p. 56
Problems	p. 58
Generating Functions of Several Variables	p. 59
The Pascal triangle	p. 59
Exponential generating functions	p. 61
The Dyck triangle	p. 63
The Bernoulli-Euler triangle and enumeration of snakes	p. 64
Representing generating functions as continued fractions	p. 72
The Euler numbers in the triangle with multiplicities	p. 78
Congruences in integer sequences	p. 79
How to solve ordinary differential equations in generating functions	p. 82
Problems	p. 83
Partitions and Decompositions	p. 87
Partitions and decompositions	p. 87
The Euler identity	p. 92
Set partitions and continued fractions	p. 95
Problems	p. 98
Dirichlet Generating Functions and the Inclusion-Exclusion Principle	p. 101
The inclusion-exclusion principle	p. 101
Dirichlet generating functions and operations with them	p. 104
Mobius inversion	p. 107
Multiplicative sequences	p. 109
Problems	p. 110
Enumeration of Embedded Graphs	p. 113
Enumeration of marked trees	p. 113
Generating functions for non-marked, marked, ordered, and cyclically ordered objects	p. 119
Enumeration of plane and binary trees	p. 120
Graph embeddings into surfaces	p. 122
On the number of gluings of a polygon	p. 132
Proof of the Harer-Zagier theorem	p. 136
Problems	p. 140
Final and Bibliographical Remarks	p. 143
Bibliography	p. 145
Index	p. 147
Table of Contents provided by Ingram. All Rights Reserved.

Lectures on Generating Functions

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