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)
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.
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.
Garrett, Ismail, and Stanton gave a general formula that contains the Rogers– Ramanujan identities as special cases. The theory of associated orthogonal polynomials is then used to explain determinants that Schur introduced in 1917 and show that the Rogers–Ramanujan identities imply the Garrett, Ismail, and Stanton seemingly more general formula. Using a… (More)
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.
To Anders Björner on his 60 th birthday. Abstract. We start with a (q, t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q, t)-binomial coefficients and… (More)
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)