How to Count: An Introduction to Combinatorics, Second Edition

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Features

  • Explains how to solve various combinatorial problems, from Euler’s problem of polygon division, Fibonacci numbers, and the Königsberg bridges problem to the four-color problem, labeled trees, and card shuffling
  • Uses problems to introduce the theory
  • Contains enough material for a short course on graph theory
  • Presents proofs of key results as well as numerous worked examples
  • Includes paired exercises, along with a full solution to one of the exercises in each pair
  • Lists suggestions for further reading

Solutions manual available for qualifying instructors

Summary

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?
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

Author Bio(s)

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.

Editorial Reviews

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