This book introduces readers to the language of generating functions, which nowadays, is the main language of enumerative combinatorics. The book 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 the exclusion-inclusion principle. In the final chapter, the author describes applications to enumeration of trees, plane graphs, and graphs embedded in two-dimensional surfaces.
Throughout the book, the author motivates readers by giving interesting examples rather than general theories. It contains numerous exercises to help students master the material. The only prerequisite is a standard calculus course. The book is an excellent text for a one-semester undergraduate course in combinatorics.
S. K. Lando, Independent University of Moscow, Moscow, Russia
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