This is a concise introduction to combinatorics with a good collection of problems. This Dover edition is an unaltered reprint of the 1958 Wiley edition, with an errata sheet.
This was one of the first textbooks of modern combinatorics, and though only about one-quarter the size of modern textbooks it covers the most important parts of the subject. The first three chapters go over basic counting techniques based on permutations and combinations and the inclusion-exclusion principle. Each remaining chapter focuses on one class of combinatoric problem, for example partitions or counting trees. Sometimes general techniques appear in these later chapters: for example, Pólya’s theory of counting is introduced in connection with counting trees. Generating series are used throughout to derive and to represent the results. Many of the results are tied to results in probability.
The problems are especially good. There are lots of them, and all are all of intermediate difficulty: requiring some ingenuity and not being an immediate application of already-covered material, but easy enough that students can be expected to work many of them. The Preface states that the problem section “is intended to carry on the development of the text, and so extend the scope of the book in a way which would have been impossible otherwise. So far as possible, the problems are put in a form to aid rather than baffle the reader.” The problems usually have some background or a sketch of the solution to help the solver get started.
In modern curricula this book would be aimed at the upper undergraduate level and most of the book could be covered in a one-semester course. There are a number of asides or problems referring to more advanced mathematics, such as the Stieltjes integral, Hermite and Legendre polynomials, and some number theory, but these are not in the main narrative and students will not be handicapped if they are not familiar with them.
Riordan’s text is old, but it still gives you a good introduction to combinatorics, is not intimidatingly thick, and is available at a bargain price as a Dover reprint. If you want something more modern, look at van Lint & Wilson’s A Course in Combinatorics. This is pitched at the same level as Riordan, is much more comprehensive and has many recent results, but fewer problems. Neither of these texts has any applications, and a good book for supplemental reading is Roberts & Tesman’s Applied Combinatorics.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.