The Matrix Ansatz, Orthogonal Polynomials, and Permutations

In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We illustrate this approach with applications to moments of orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for Dennis Stanto

Set-polynomials and polynomial extension of the Hales-Jewett Theorem

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established: Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a \subset V and a nonempty set \gamma \subseteq {1,...,N} such that a \cap (\gamma^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (\gamma^{d} x {1}), a \cup (\gamma^{d} x {2}), ..., a \cup (\gamma^{d} x {q}) are all of the same color. This ``polynomial'' Hales-Jewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.Comment: 43 pages, published versio