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 phenonmenon of Reiner, Stanton and White. The main tool is a class of symmetric functions recently introduced in work of two of the authors.
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. It was partly motivated as an extension of the q=-1 phenomenon introduced earlier by Stembridge. We give a survey of the current literature on cyclic sieving. The techniques used to prove that the phenomenon holds are discussed, including ideas from representation theory such as Springer's Theorem on regular elements in complex reflection groups.
Let Πn denote the set of all set partitions of {1,2,...,n}. We consider two subsets of Πn, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let En⊆Πn be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, Tn-1. Given π∈Πm and σ∈Πn, define their slash product to be π|σ=π∪(σ+m)∈Πm+n where σ+m is the partition obtained by adding m to every element of every block of σ. Call a partition atomic if it can not be written as a nontrivial slash product and let An⊆Πn denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of NCSym, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, En=An for all n≥0. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to NCSym. We end with some remarks and an open problem.