2nd Edition

How to Count An Introduction to Combinatorics, Second Edition

By R.B.J.T. Allenby, Alan Slomson Copyright 2011
    446 Pages 164 B/W Illustrations
    by Chapman & Hall

    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.

    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
    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
    A Formula for the Stirling Numbers

    Stirling and Catalan Numbers
    Stirling Numbers
    Permutations and Stirling Numbers
    Catalan Numbers

    Partitions and Dot Diagrams
    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

    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
    The Axioms for Group Actions

    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



    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