## Recent papers

The cyclic sieving phenomenon (with V. Reiner and D. Stanton), Journal of Combinatorial Theory A, 2004, (pdf)

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

Mahonian Z statistics (with Jennifer Galovich), Discrete Math, to appear, (pdf)

Abstract The $Z$ statistic of Zeilberger and Bressoud is computed by summing the major index of the 2-letter subwords. We generalize this idea to other splittable Mahonian statistics. We call splittable Mahonian statistics which produce other splittable Mahonian statistics in this fashion {\it $Z$-Mahonian}. We characterize $Z$-Mahonian statistics and include several examples.

The Schur cone and the cone of log concavity, (pdf)

Abstract Let $\{h_1,h_2,\dots\}$ be a set of algebraically independent variables. We ask which vectors are extreme in the cone generated by $h_ih_j-h_{i+1}h_{j-1}$ ($i\geq j>0$) and $h_i$ ($i>0$). We call this cone the {\it cone of log concavity}. More generally, we ask which vectors are extreme in the cone generated by Schur functions of partitions with $k$ or fewer parts. We give a conjecture for which vectors are extreme in the cone of log concavity. We prove the characterization in one direction and give partial results in the other direction.