●Preface to the English Edition
Preface
Chapter 1.Formal Power Series and Generating Functions. Operations with Formal Power Series.Elementary Generating Functions
§1.1.The lucky tickets problem
§1.2.First conclusions
§1.3.Generating functions and operations with them
§1.4.Elementary generating functions
§1.5.Diferentiating and integrating generating functions
§1.6.The algebra and the topology of formal power series
§1.7.Problems
Chapter 2.Generating Functions for Well-known Sequences
§2.1.Geometric series
§2.2.The Fibonacci sequence
§2.3.Recurrence relations and rational generating functions
§2.4.The Hadamard product of generating functions
§2.5.Catalan numbers
§2.6.Problems
Chapter 3.Unambiguous Formal Grammars. The Lagrange Theorem
§3.1.The Dyck Language
§3.2.Productions in the Dyck language
§3.3.Unambiguous formal grammars
§3.4.The Lagrange equation and the Lagrange theorem
§3.5.Problems
Chapter 4.Analytic Properties of Functions Represented as Power Series and the Asymptotics of their Coefficients
§4.1.Exponential estimates for asymptotics
§4.2.Asymptotics of hypergeometric sequences
§4.3.Asymptotics of coefficients of functions related by the Lagrange equation
§4.4.Asymptotics of coeficients of generating series and singularities on the boundary of the disc of convergence
§4.5.Problems
Chapter 5.Generating Functions of Several Variables
§5.1.The Pascal triangle
§5.2.Exponential generating functions
§5.3.The Dyck triangle
§5.4.The Bernoulli-Euler triangle and enumeration of snakes
§5.5.Representing generating functions as continued fractions
§5.6.The Euler numbers in the triangle with multiplicities
§5.7.Congruences in integer sequences
§5.8.How to solve ordinary differential equations in generating functions
§5.9.Problems
Chapter 6.Partitions and Decompositions
§6.1.Partitions and decompositions
§6.2.The Euler identity
§6.3.Set partitions and continued fractions
§6.4.Problems
Chapter 7.Dirichlet Generating Functions and the Inclusion-Exclusion Principle
§7.1.The inclusion-exclusion principle
§7.2.Dirichlet generating functions and operations with them
§7.3.Mobius inversion
§7.4. ltiplicative sequences
§7.5.Problems
Chapter 8.Enumeration of Embedded Graphs
§8.1.Enumeration of marked trees
§8.2.Generating functions for non-marked, marked. ordered, and cyclically ordered objects
§8.3.Enumeration of plane and binary trees
§8.4.Graph embeddings into surfaces
§8.5.On the number of gluings of a polygon
§8.6.Proof of the Harer-Zagier theorem
§8.7.Problems
Final and Bibliographical Remarks
Bibliography
Index

本書向讀者介紹了生成函數的語言，它是當今計數組合學的主要語言。本書從定義、簡單的屬性和許多生成函數的例子開始，之後討論了形式語法、多變量生成函數、分拆和分解以及容斥原理等主題。在最後一章中，作者描述了樹、平面圖和嵌入在二維曲面中的圖的計數應用。在全書中，作者通過提供有趣的例子而不是一般理論來激發讀者的興趣。本書包含許多練習來幫助學生學習。閱讀本書隻需讀者先修過一門標準的微積分課程。本書是一學期的組合數學本科課程的優秀教材。