The Matrix Ansatz, orthogonal polynomials, and permutations

Abstract

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.To Dennis Stanton with admiration.where Z n = W |(D + E) n |V .1991 Mathematics Subject Classification. 05A15,05A19,33C45.

Crossings of signed permutations and $q$-Eulerian numbers of type $B$

“…The type B case was studied in [9,7]. In particular, the crossings of signed permutations were defined in [7]. In this article, we show how this definition give nice q-Eulerian numbers of type B, which are a type B analogue of the E n,k (q) mentioned above.…”

“…Lam and Williams [21], in their work on the totally positive Grassmanian cells, defined analogues of permutation tableaux for other groups. The type B case was studied in [9,7]. In particular, the crossings of signed permutations were defined in [7].…”

“…The notion of crossing of signed permutations was first considered in [7]. Now let k and n be integers with 0 ≤ k ≤ n. The type B Eulerian number E B n,k is the number of π ∈ B n with des B (π) = k, see [4].…”

“…Corteel et al [7] showed that B n (1, t, q) is a generating function for type B permutation tableaux. These tableaux were introduced by Lam and Williams [21].…”

“…The purpose of this article is to study the type B analogue, jointly with some excedance statistics on signed permutations, and more generally flag excedances. This work is motivated by permutation tableaux of type B introduced by Lam and Williams [21], and natural statistics that can be read on these tableaux [7]. We thus obtain q-Eulerian numbers of type B, or more generally flag q-Eulerian numbers of type B, and investigate their combinatorial properties.…”