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How to Count: An Introduction to Combinatorics

R. B. J. T. Allenby and Alan Slomson
Chapman & Hill/CRC
Publication Date: 
Number of Pages: 
Discrete Mathematics and Its Applications
[Reviewed by
Alex Bogomolny
, on

Joint review of

  • How To Count: An Introduction to Combinatorics, by R. B. J. T. Allenby and Alan Slomson
  • Introduction to Combinatorics, W. D. Wallis and J. G. George

The two books under review are introductory combinatorics texts, also suitable for individual study. I shall refer to them by the combination of the first letters of the authors’ last names: AS and WG. Both book are thoughtfully written, contain plenty of material and exercises. Understandably, there is a considerable overlap. Common to the books are the essential topics, such as permutations and combinations, the Inclusion-Exclusion principle, generating functions and recurrence relations, graphs and graph algorithms, groups of permutations, counting patterns, Pólya counting, the pigeonhole principle and Ramsey theory, and Catalan and Sterling numbers.

AS features chapters on integer partitions, barely touched in WG, and, in general, covers more expansively the common topics, especially, group and Ramsey theories, matching and marriages. WG, on the other hand, devotes separate chapters to coding theory, Latin squares, balanced incomplete block designs, and applications of linear algebra to combinatorics. AS (p. 228) observes that “Latin squares are interesting and important combinatorial objects, but because of shortage of space we are not able to discuss them in this book.” There are briefer, say section-level, differences: rook polynomials, Euclidean Ramsey theory (AS), sum-free sets, exponential generating functions (WG).

In both books exercises come in two categories: A and B. The A exercises are supplied with solutions, the B exercises (which in WG are collected in separate sections) are not. WG also offers lists of problems — generally more sophisticated questions — that are accompanied by hints and solutions at the end of the book.

Both books begin with extensive Introductions that outline the contents by posing questions and problems that the books deal with subsequently. AS offers short biographical information in footnotes; WG in a separate Appendix. In addition, WG covers some background information (proof techniques, matrices and vectors) in two separate appendices.

WG provides numerous fragments of Mathematica code and this is a nice touch. For example, on pp. 259–260 the authors discuss the problem of determining the number of ways to cover a 2×6 board with nonoverlapping dominos. They form a digraph and its adjacency matrix A and then rely on Mathematica’s

B = Inverse[IdentityMatrix[6] - x*A];

to obtain the generating function

(1 – 2x2 + x4)/(1- x – 6x2 – x3 + 6x4 + 2x5 – x6)

which they then expand into a power series with


and subsequently elucidate the meanings of every coefficient.

For a balance check, AS goes to a considerably greater length introducing elements of group theory as a tool towards Frobenius and Pólya counting theorems.


At this point I would like to confess that the two books have been sitting on my desk for a while now. Somehow my first impression was so distressing that I could not see the point of reviewing them. As luck would have it, I opened the books the first time on pages with misprints. For AS, it was a confusing swap of “>” for “<” on page 296. For WG, it was a section titled “Applications of P(n, k) and C(n, k)” that had no mention of P(n, k).

Truth be told, I actually came across only a few more typos in AS and eventually found the text very readable and useful. I can’t say the same of WG. The number of typos there is bewildering. For example, on pages 196–197 there are three:

  1. B1A4B3B4B5B6B7 instead of B1B2B3B4B5B6B7
  2. ainC instead of a∈C, and
  3. N instead of n.

These were actually trifles. On page 73, Example 4.3 refers to Example 2.2 (p. 27), while the reference should be to Example 2.6 (p. 31) The diagram in the latter example is exactly the same as that accompanying Example 4.3, even though the diagram is not quite suitable for the earlier stage.

I truly came to grief with the WG’s index. None of the terms I wanted to check could be found on the pages the Index pointed me to. Only after I decided to record the discrepancies, I realized that a major part of the index was off by 2 pages. It’s not that, bad one may say, but not every student will be as savvy or as determined to discover the truth. Further, the Index contains the word “Arrangement”, but it is not defined (far as I can see) anywhere in the book; what the index poinst to is the definition of “Derangement”.

To give one more example, Theorem 13.3 reads, “The balanced bipartite graph B has exactly Per(B(G)) perfect matchings.” As a matter of fact the term “balanced bipartite graph” is not found in the index and, as far as I can judge, is not defined anywhere in the book, at least not where I would expect it to be. The same holds for the “Cancellation law” entry.


Thus, for a long time I was at a loss whether to write a review or to let time pass. Then somebody tweeted me a review (by John Trimble) of Writing With Style: Conversations on the Art of Writing, a book by James M. Lang. The review is full of praise, in particular from the book’s current editor Brad Potthoff:

He brings out the best in his editors and asks as much of them as he does of himself. As such, working with John is a joyous challenge and one of the highlights of my career.

This note gave me a point of reference and an impulse to pull myself together. Whether or not this is due to the authors’ failure to engage the editors at CRC, the fact is that the latter did a lousy job editing the books, crucially so for WG.

Alex Bogomolny is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math.

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