The study of integer partitions is at once both incredibly simple and incredibly complex. The basic question is very easy: how many ways can one decompose an integer additively into other positive integers? For example, 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so there are a total of 7 different partitionings of the number 5. However, a general formula for the number of partitions of n is not known, and in fact it is not even known how to tell whether n will have an even or an odd number of partitions. Partitions were first studied by Euler, and have been seriously studied by some of the biggest names in combinatorics and number theory, including Hardy, Legendre, Ramanujan, Rademacher, Dyson, Wilf, Ono, and many others, who have found connections between Partition theory and modular forms, quantum physics, and throughout mathematics.
Since its publication in the 1970's, George Andrews' book The Theory Of Partitions has become the standard reference for the subject, and remains one of the best introductions to the subject for mathematicians. It is not for amateurs, however, despite the fact that one of the appeals of the definition of partitions lies in its simplicity. In fact, until now there has not been a good introduction to the subject at an elementary level. Integer Partitions, by George Andrews and Kimmo Eriksson, is written at a level that most undergraduates, and even many high school students, could follow quite easily. The book starts from the very beginnings of the subject, and introduces the math necessary to understand Euler's Identity (which says that the number of partitions into odd parts is the same as the number of partitions into distinct parts), the more complicated Rogers-Ramanujan Identities, generating functions, and a number of other topics in this circle of ideas. The exposition is clear, and rather than dwelling on lots of technical details, many of the proofs can be found in the exercises with copious hints. The authors intended to write a book which exposes the reader to lots of ideas without going into tremendous depth on any of them, and in this they were successful.
To me, the book's major failing comes when it tries to hide many of the details and the greater mathematical context. The authors often refer the reader to other books and papers for further details, and more than a few times I found myself wanting to go track down these references before I could continue reading their book. While I understand that the authors did not intend to write an encyclopedia on the subject and instead to write an introduction to this vast area, they went too far in that direction for my tastes. Furthermore, the stated goal of this book is to be used in an undergraduate topics course in Partition Theory, but most of my students would lose interest in the topic before the semester ran out unless they saw the larger context and some of the applications of the theory that Andrews and Eriksson hide in the bibliography and exercises. However, these were conscious decisions on the part of the authors, and while I may have made different decisions, I certainly think the authors did an admirable job of writing the book that they wanted to write.
Darren Glass is Assistant Professor of Mathematics at Columbia University.