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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.
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)
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)