How to Count : An Introduction to Combinatorics, Second Edition book cover
2nd Edition

How to Count
An Introduction to Combinatorics, Second Edition

ISBN 9781420082609
Published August 12, 2010 by Chapman & Hall
444 Pages 164 B/W Illustrations

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Book Description

Emphasizes a Problem Solving Approach
A first course in combinatorics

Completely revised, How to Count: An Introduction to Combinatorics, Second Edition shows how to solve numerous classic and other interesting combinatorial problems. The authors take an easily accessible approach that introduces problems before leading into the theory involved. Although the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics.

New to the Second Edition
This second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet’s pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises.

Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Pólya’s counting theorem.

Table of Contents

What’s It All About?. Permutations and Combinations. Occupancy Problems. The Inclusion-Exclusion Principle. Stirling and Catalan Numbers. Partitions and Dot Diagrams. Generating Functions and Recurrence Relations. Partitions and Generating Functions. Introduction to Graphs. Trees. Groups of Permutations. Group Actions. Counting Patterns. Pólya Counting. Dirichlet’s Pigeonhole Principle. Ramsey Theory. Rook Polynomials and Matchings. Solutions to the A Exercises. Books for Further Reading. Index.

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Alan Slomson taught mathematics at the University of Leeds from 1967 to 2008. He is currently the secretary of the United Kingdom Mathematics Trust.

R.B.J.T. Allenby taught mathematics at the University of Leeds from 1965 to 2007.


The current edition is about 60% longer and represents an extensively updated collaboration coauthored with R.B.J.T. Allenby. Both authors have decades of experience teaching related material at the University of Leeds. …
The book is beautifully structured to facilitate both instruction in a classroom as well as self-instruction. … Every section of the book has a number of [paired] exercises which are designed to solidify and build understanding of the topics in the section. … This is exactly the kind of exercise regimen serious readers, instructors, and students need and are so rarely provided. …
Another pedagogical asset of the text is the extensive incorporation of historical anecdotes about the discoverers of the results. … it fosters an admiration of the developers of the field, an attitude which is key to transforming students of mathematics into professional mathematicians. …
The authors have created an interesting, instructive, and remarkably usable text. The book clearly benefits instructors who need a solid, readable text for a course on discrete mathematics and counting. In fact, for any professional who wants an understandable text from which they can acquire a broad and mathematically solid view of many of the classic problems and results in counting theory, including their origin, proof, and application to other problems in combinatorics, this book is recommended.
—James A. McHugh, SIAM Review, 54 (1), 2012

… thoughtfully written, contain[s] plenty of material and exercises … very readable and useful …
MAA Reviews, February 2011

The reasons I adopted this book are simple: it’s the best one-volume book on combinatorics for undergraduates. It begins slowly and gently, but does not avoid subtleties or difficulties. It includes the right mixture of topics without bloat, and always with an eye to good mathematical taste and coherence. Enumerative combinatorics is developed rather fully, through Stirling and Catalan numbers, for example, before generating functions are introduced. Thus this tool is very much appreciated and its ‘naturalness’ is easier to comprehend. Likewise, partitions are introduced in the absence of generating functions, and then later generating functions are applied to them: again, a wise pedagogical move. The ordering of chapters is nicely set up for two different single-semester courses: one that uses more algebra, culminating in Polya’s counting theorem; the other concentrating on graph theory, ending with a variety of Ramsey theory topics. … I was very much impressed with the first edition when I encountered it in 1994. I like the second edition even more. …
—Paul Zeitz, University of San Francisco, California, USA

Completely revised, the book shows how to solve numerous classic and other interesting combinatorial problems. … The reading list at the end of the book gives direction to exploring more complicated counting problems as well as other areas of combinatorics.
Zentralblatt MATH 1197