This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

‘Proofs and Confirmations is one of the most brilliant examples of mathematical exposition that I have encountered, in many years of reading the same. This is not for the faint-hearted, nor is Proofs and Confirmations a book that can be read in an easy chair, like a novel; it demands active participation by the reader. But Bressoud rewards such readers with a panorama of combinatorics today and with renewed awe at the human ability to penetrate the deeply hidden mysteries of pure mathematics.’

Herbert S. Wilf Source: Science

‘The unexpected twists and turns will hardly be matched in any novel - this book allows us all to share in the excitement … a brilliant book.’

Alun O. Morris

‘I strongly recommend the book as an account of a remarkable mathematical development.’

P. J. Cameron Source: Proceedings of the Edinburgh Mathematical Society

‘This is an excellent book which can be recommended without hesitation, not only to specialists in the field, but to any mathematician with time to read something interesting and nicely written.’

Source: EMS