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.

The Second Waterloo Workshop on Computer Algebra (WWCA 2008) was held May 5-7, 2008 at Wilfrid Laurier University, Waterloo, Canada. This conference was dedicated to the 70th birthday of Georgy Egorychev (Krasnoyarsk, Russia), who is well known and highly regarded as the author of the influential, milestone book "Integral Representation and the Computation of Combinatorial Sums," which described a regular approach to combinatorial summation, today also known as the method of coefficients. Another great success of this Russian mathematician came in 1980, when he solved the van der Waerden conjecture on the determination of the minimum of the permanent of a doubly stochastic matrix and was awarded the D. R. Fulkerson Prize.

This book presents a collection of selected formally refereed papers submitted after the workshop. The topics discussed in this book are closely related to Georgy Egorychev’s influential works.