This book is a comprehensive treatment of the combinatorics of counting, which is related to the analysis of algorithms and intimately connected with probability. The book takes an entirely combinatorial perspective, going through basic counting arguments (such as counting the number of ways of arranging characters, items, and sets), combinatorial identities, formal power series, and symmetric polynomials. The book would be suitable for advanced undergraduates or as a graduate text. It would also be a good book for a computer scientist who wants to learn about enumeration.

The book begins with an enchanting introductory chapter. It details a series of interesting motivational problems, designed to whet the reader’s appetite and demonstrate the breadth of the subject matter.

Chapter 1 is an introduction to (or a review of) basic counting. It covers the well-known sum and product rules, as well as functions (injections, surjections, and bijections). It includes a plethora of examples that demonstrate the principles and keep the reader’s interest. For example, it offers a detailed analysis of the special card hands in poker.

Chapter 2 introduces combinatorial identities and recursions. These are largely binomial and factorial results, and here, the author also introduces the idea of a combinatorial, or bijective, proof, demonstrating these techniques on combinatorial objects such as lattice paths, integer partitions and diagrams, and set partitions.

Chapter 3 focuses on counting problems in graph theory, looking at vertex degrees, permutations, trees, and chromatic polynomials. Later chapters do not depend on this one, and it could be omitted from a course that does not consider graph problems.

Chapter 4 covers the ins and outs of the principle of inclusion-exclusion, giving various proofs of it, a simplification, and a number of applications that revisit earlier elements such as Stirling numbers and chromatic polynomials. This chapter also covers Möbius inversion and posets.

Chapter 5, “Ranking and Unranking,” primarily concerns algorithms for ranking and generating combinatorial objects such as words, permutations, and subsets. Counting books do not always include this topic; texts on combinatorial algorithms more commonly cover it, and it is nice to see it included here.

Chapter 6 marks the move into more algebraic topics, and also marks the first time that generating functions are introduced as a technique for counting weighted combinatorial objects. The primary vehicle for demonstrating these ideas is the q-binomial coefficient (here called the “quantum binomial coefficient”); the author derives quantum versions of several identities.

Chapters 7 through 9 are highly technical algebraic chapters that cover the topics of formal power series and group actions. Chapter 10 transitions to specialized topics, concentrating on a combinatorial approach to Schur functions and related concepts, including tableaux.

Chapter 11 contains more combinatorial results for symmetric polynomials, and introduces the classical topics of skew Schur functions, the Jacobi-Trudi identity, and the Littlewood-Richardson rule.

Chapter 12 finishes the book with additional topics: the Chung-Feller theorem, parking functions, tournaments, the hook-length formula, and Pfaffians, among other topics.

The book is clearly written and packed with examples and problems (it provides answers and hints to selected problems at the end). The organization is superior, with helpful tables of notation and definitions at the end of each chapter, along with a point-by-point summary of the chief topics.

The main quibble with the book is the role that it gives to generating functions. The book does not introduce them until chapter 6, and exponential generating functions play a small part overall. It would be useful to have them appear earlier, since the framework and philosophy of generating functions help organize one’s thinking about enumerative problems and provide an approach for tackling almost any new problem. Aside from this philosophical difference on generating functions, however, the book is an excellent one, and is a comprehensive and welcome addition to the area.