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How to Count
An Introduction to Combinatorics, Second Edition
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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 selfstudy, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusionexclusion 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 InclusionExclusion 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 PictureFree Definition
Isomorphism of Graphs
Paths and Connected Graphs
Planar Graphs
Eulerian Graphs
Hamiltonian Graphs
The FourColor Theorem
Trees
What Is a Tree?
Labeled Trees
Spanning Trees and Minimal Connectors
The ShortestPath 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(s)
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.
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 selfinstruction. … 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 onevolume 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 singlesemester 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 MATH 1197
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