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Suitable for self-study or a first course in combinatorics at the undergraduate level, How to Count: An Introduction to Combinatorics, Second Edition follows a similar approach to its predecessor. This second edition continues to focus on counting problems and emphasize a problem solving approach. It includes a new chapter on graph theory and many more exercises, some with full solutions or hints. The authors provide proofs of all significant results and illustrate applications from other areas of mathematics, such as elementary ideas from analysis in proving Stirling's formula. A solutions manual is available for qualifying instructors.

What's It All About? What Is Combinatorics? Classic Problems What You Need to Know Are You Sitting Comfortably? Permutations and Combinations The Combinatorial Approach Permutations CombinationsApplications to Probability Problems The Multinomial Theorem Permutations and Cycles Occupancy Problems Counting the Solutions of Equations New Problems from Old A Reduction Theorem for the Stirling Numbers The Inclusion-Exclusion Principle Double Counting Derangements A Formula for the Stirling Numbers Stirling and Catalan Numbers Stirling Numbers Permutations and Stirling Numbers Catalan Numbers Partitions and Dot Diagrams Partitions Dot Diagrams A Bit of Speculation More Proofs Using Dot Diagrams Generating Functions and Recurrence Relations Functions and Power Series Generating Functions What Is a Recurrence Relation? Fibonacci Numbers Solving Homogeneous Linear Recurrence Relations Nonhomogeneous Linear Recurrence Relations The Theory of Linear Recurrence Relations Some Nonlinear Recurrence Relations Partitions and Generating Functions The Generating Function for the Partition Numbers A Quick(ish) Way of Finding p(n) An Upper Bound for the Partition Numbers The Hardy-Ramanujan Formula The Story of Hardy and Ramanujan Introduction to Graphs Graphs and Pictures Graphs: A Picture-Free Definition Isomorphism of Graphs Paths and Connected Graphs Planar Graphs Eulerian Graphs Hamiltonian Graphs The Four-Color Theorem Trees What Is a Tree? Labeled Trees Spanning Trees and Minimal Connectors The Shortest-Path Problem Groups of Permutations Permutations as Groups Symmetry Groups Subgroups and Lagrange's Theorem Orders of Group Elements The Orders of Permutations Group Actions Colorings The Axioms for Group Actions Orbits Stabilizers Counting Patterns Frobenius's Counting Theorem Applications of Frobenius's Counting Theorem Polya Counting Colorings and Group Actions Pattern Inventories The Cycle Index of a Group Polya's Counting Theorem: Statement and Examples Polya's Counting Theorem: The Proof Counting Simple Graphs Dirichlet's Pigeonhole Principle The Origin of the Principle The Pigeonhole Principle More Applications of the Pigeonhole Principle Ramsey Theory What Is Ramsey's Theorem? Three Lovely Theorems Graphs of Many Colors Euclidean Ramsey Theory Rook Polynomials and Matchings How Rook Polynomials Are Defined Matchings and Marriages Solutions to the A Exercises Books for Further Reading Index

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.

! 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