Lectures on Generating Functions
About this Title
S. K. Lando, Independent University of Moscow, Moscow, Russia. Translated by Dr Sergei Lando, Independent University of Moscow, Moscow, Russia
Publication: The Student Mathematical Library
Publication Year 2003: Volume 23
ISBNs: 978-0-8218-3481-7 (print); 978-1-4704-1819-9 (online)
MathSciNet review: MR2013270
MSC: Primary 05-01; Secondary 05A15, 33C90
In combinatorics, one often considers the process of enumerating objects of a certain nature, which results in a sequence of positive integers. With each such sequence, one can associate a generating function, whose properties tell us a lot about the nature of the objects being enumerated. Nowadays, the language of generating functions is the main language of enumerative combinatorics.
This book is based on the course given by the author at the College of Mathematics of the Independent University of Moscow. It starts with definitions, simple properties, and numerous examples of generating functions. It then discusses various topics, such as formal grammars, generating functions in several variables, partitions and decompositions, and the exclusion-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 also contains 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.
Advanced undergraduates, graduate students, and research mathematicians interested in modern methods of combinatorics.
Table of Contents
- Chapter 1. Formal power series and generating functions. Operations with formal power series. Elementary generating functions
- Chapter 2. Generating functions for well-known sequences
- Chapter 3. Unambiguous formal grammars. The Lagrange theorem
- Chapter 4. Analytic properties of functions represented as power series and their asymptotics of their coefficients
- Chapter 5. Generating functions of several variables
- Chapter 6. Partitions and decompositions
- Chapter 7. Dirichlet generating functions and the inclusion-exclusion principle
- Chapter 8. Enumeration of embedded graphs
- Final and bibliographical remarks