Dennis Stanton

Cranks andt-cores

SummaryNew statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujans congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to thet-core of a partition. Usingq-series, some explicit formulas are given for the number of partitions which aret-cores. Some related questions for self-conjugate and distinct partitions are discussed.

The cyclic sieving phenomenon

The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridges q = -1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Polya-Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field q-analogues.

The combinatorics of q -Hermite polynomials and the Askey-Wilson integral

The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilsons 4φ3 polynomials. A special case of this result gives the linearization formula for q-Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q-Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q-analogue of Mehlers formula.

Strange Evaluations of Hypergeometric Series

Many evaluations of terminating hypergeometric series at arguments other than 1 are given. Some are equivalent to some unpublished work of Gosper, while others are new. In particular, two new evaluations of

Group actions on Stanley-Reisner rings and invariants of permutation groups

{}_7 F_6

Orthogonal Polynomials and Chevalley Groups

’s with four parameters are stated. The main technique is a change of variables formula which is equivalent to the Lagrange inversion formula. A new proof of Whipple’s transformation of a very well poised

q -Taylor theorems, polynomial expansions, and interpolation of entire functions

{}_7 F_6

A Convolution Formula for the Tutte Polynomial

into a Saalschutzian

Classical orthogonal polynomials as moments

{}_4 F_3

Octobasic Laguerre polynomials and permutation statistics

is a corollary.