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.

Readership

Advanced undergraduates, graduate students, and research mathematicians interested in modern methods of combinatorics.

Reviews

"A crisp and sophisticated text ... More examples than general theory. Covers standard material, but digs deeper ... An enjoyable read for professionals."

-- MAA Monthly

"(This book) is driven by very, very interesting problems and examples."

-- MAA Reviews

Table of Contents

• Formal power series and generating functions. Operations with formal power series. Elementary generating functions
• Generating functions for well-known sequences
• Unambiguous formal grammars. The Lagrange theorem
• Analytic properties of functions represented as power series and their asymptotics of their coefficients
• Generating functions of several variables
• Partitions and decompositions
• Dirichlet generating functions and the inclusion-exclusion principle
• Enumeration of embedded graphs
• Final and bibliographical remarks
• Bibliography
• Index