The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge's q ¼ À1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Poíya–Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite… (More)
We explain how the moments of the (weight function of the) Askey Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors [11, 12]. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant than the formula given in . Then we use techniques… (More)
Necessary and sufficient conditions are given for an s-block of integer partitions to be contained in a t-block. The generating function for such partitions is found analytically, and also bijectively, using the notion of an (s, t)-abacus. The largest partition which is both an score and a t-core is explicitly given.
We evaluate several integrals involving generating functions of continuous q-Hermite polynomials in two diierent ways. The resulting identities give new proofs and generalizations of the Rogers-Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers-Ramanujan identities without the Jacobi triple product… (More)
Young's lattice of a partition λ consists of all partitions whose Ferrers diagrams fit inside λ. Several infinite families of partitions are given whose Young's lattice is not rank unimodal. Some related problems are discussed.
An introduction is given to the use orthogonal polynomials in distance regular graphs and enumeration. Some examples in each area are given, along with open problems.
Let M be a finite matroid with rank function r. We will write A ⊆ M when we mean that A is a subset of the ground set of M , and write M | A and M/A for the matroids obtained by restricting M to A, and contracting M on A respectively. Let M * denote the dual matroid to M. (See  for definitions). The main theorem is Theorem 1. The Tutte polynomial T M (x,… (More)