2nd Edition

# How to Count An Introduction to Combinatorics, Second Edition

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*Emphasizes a Problem Solving ApproachA 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.

**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

Combinations

Applications 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

**Pólya Counting **Colorings and Group Actions

Pattern Inventories

The Cycle Index of a Group

Pólya’s Counting Theorem: Statement and Examples

Pólya’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**

### Biography

**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 2011The 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, USACompletely 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 MATH1197