Enumerative combinatorics--a subfield of combinatorics--is concerned with the number of ways patterns can be formed. This book on enumerative combinatorics is the revised second edition of a book that first appeared in 1986 [1]. The book is a masterpiece on the subject. It is a scholarly work by the erudite professor R. P. Stanley, who is universally accepted as a leading expert in the field of combinatorics and its applications. Numerous exercises, graded according to their level of difficulty, make this book suitable for amateurs and experts alike. The mathematical prerequisites for understanding the material are minimal; however, novices will need to put in some hard work. This book is reasonably priced despite the wealth of information it contains.

In the first of four sections, the author offers an introduction to enumerative combinatorics. The second section acquaints the reader with sieve methods. The third section focuses on partially ordered sets. The fourth section deals with rational generating functions. The style of exposition in the book should be praised. The author provides numerous all-inclusive frameworks for solving enumeration problems.

Combinatorics often seems like a motley assortment of topics even to mathematicians. However, the author successfully presents it as a unified whole. The book is simply magnificent in the way it presents difficult mathematics and includes many fascinating exercises. The original edition [1] became a standard guide to enumerative combinatorics for beginners as well as experts. This second edition has been thoroughly revised and now contains ten new topics as well as over 350 new exercises, many with solutions. The changes reflect the developments that have taken place since 1986, and the references have been updated.

Today, few mathematicians are willing (and even fewer are able) to write in a manner that will be beneficial for those who are not proficient in their domain. This book and its companion volume [2] are chefs-d’oeuvre of mathematical authorship. The book contains an enormous amount of information that is readable as well as worthy of being taught. There is plenty of practice for readers thanks to the many exercises. The very elaborate bibliographies and notes at the end of the sections guide the interested reader in discovering even more than what is covered in the book.

On the negative side, the second edition of the book still has typographical errors that could have been avoided; they are listed on the author’s Web site (http://www-math.mit.edu/~rstan/ec/). I would also have been happier had a slightly bigger font been used. Nonetheless, this book is a definitive text on enumerative combinatorics. I strongly recommend it as a bible on enumerative combinatorics for mathematicians. Needless to say, both volumes are a “must buy” for libraries.