Advances in Combinatorial Mathematics

It is a pleasure for me to have the opportunity to write the foreword to this volume, which is dedicated to Professor Georgy Egorychev on the occasion of his seventieth birthday. I have learned a great deal from his creative and important work, as has the whole world of mathematics. From his life¿s work (so far) in having made d- tinguished contributions to ?elds as diverse as the theory of permanents, Lie groups, combinatorial identities, the Jacobian conjecture, etc., let me comment on just two of the most important of his research areas. The permanent of an n×n matrix A is Per(A)= a a ...a , (1) ? 1,i 2,i n,i 1 2 n extended over the n! permutations{i ,...,i} of{1,2,...,n}. Thus, the permanent 1 n is ¿like the determinant except for dropping the sign factors from the terms.¿ H- ever by dropping those signs, one loses almost all of the friendly characteristics of determinants, such as the fact that det(AB)= det(A)det(B), the invariance under elementary row and column operations, and so forth. The permanent is a creature of multilinear algebra, rather than of linear algebra, and is much crankier to deal with in virtually all of its aspects, both theoretical and algorithmic.