Eulerian quasisymmetric functions and cyclic sieving

It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group ^Sn generated by the n-cycle (1,2,...,n) on the set of permutations of fixed cycle type and fixed number of excedances provides an instance of the cyclic sieving phenomenon of Reiner, Stanton and White. The main tool is a class of symmetric functions recently introduced in work of two of the authors.