Combinatorial Enumeration

Dover, formerly very weak in combinatorics, has recently republished two classics, MacMahon's Combinatory Analysis and Riordan's An Introduction to Combinatorial Analysis, as well as the book under review, which first appeared in 1983. Along with two French works (Berge and Comtet), these were the outstanding books on enumeration when I started graduate school in 1985. The first volume of Stanley's Enumerative Combinatorics joined their ranks a year later, and Stanley's second volume set a new standard when it finally came out in 1998.

Many mathematicians still hold the view that combinatorics is basically a collection of tricks, rather than a coherent theory. Gian-Carlo Rota, who wrote the Foreword to the book, spent much of his career trying to change both the reality and the perception of this view, and this book was also written against it. In the preface the authors say that "the examples...have been selected to illustrate the variety of the enumerative results that can be obtained from a single decomposition. Many of these can be derived separately, and more quickly, by methods peculiar to the particular problem. However...." Theirs is a systematic development of enumerative theory from the generating function point of view, at a high level with a minimum of tricks and artifices. To achieve this the authors find it necessary to use a cumbersome notation that makes the book unrewarding to browse through---one has to read it from the beginning to really appreciate what they have accomplished. There are many beautiful results here, but the beauty is often obscured by the presentation.

It is hard not to compare the book with Stanley's books, a comparison that it clearly loses. In the Foreword to Stanley's second volume Rota found it "impossible to predict when Richard Stanley's two-volume exposition of combinatorics may be superseded. No one will dare try, let alone be able, to match the thoroughness of coverage, the care for detail, the definitiveness of proof, the elegance of presentation." Goulden and Jackson are two of the outstanding enumerators of the last 30 years, and they would undoubtedly write a better book now. Presumably they chose not to do a second edition at least in part because Stanley has set the bar so high.

That said, there is still a substantial amount of material in their book that is not in Stanley, especially in chapters 2, 4 and 5. The introductory chapter 1 is a brisk and very nice development of formal power series, up through a multivariate Lagrange inversion formula. Another strength (as with Stanley's books) is an extensive collection of exercises, with solutions; the solutions occupy pages 339-541! The bibliography is also very good, even though it has not been updated. If one only bought books as good as Stanley's then one would have a very small library indeed. This book still belongs in the collection of any serious student of enumeration. At $34.95 it is unusually expensive for a Dover paperback, but I paid over twice that for it in graduate school. I am very happy to see it back in print.

Warren Johnson (warren.johnson@conncoll.edu ) is visiting assistant professor of mathematics at Connecticut College.