Permutation Statistics and the Cyclic Sieving Phenomenon
by
John Shareshian
Washington Univeristy
Coauthors: Michelle Wachs, University of Miami

Let G be a cyclic group acting on a set X and let P be a polynomial with integer coefficients. The triple (G, X, P) exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton and White, if whenever g in G has order d, the number of points in X fixed by g is obtained by substituting a primitive dth root of unity into P. There are many known interesting instances of this phenomenon, and I will present additional examples in which X is a union of conjugacy classes in the symmetric group S_n, G is a subgroup of S_n generated by an n-cycle or (n-1)-cycle acting on X by conjugation, and P is the generating polynomial for the restriction of a standard permutation statistic to X.

Date received: February 21, 2009