Eulerian quasisymmetric functions and cyclic sieving

TL;DR

This paper demonstrates that a refined q-analogue of Eulerian numbers, combined with a specific symmetric group action, exemplifies the cyclic sieving phenomenon, using novel symmetric functions as the main tool.

Contribution

It introduces a new connection between Eulerian numbers, symmetric functions, and cyclic sieving, expanding the understanding of these combinatorial structures.

Findings

Confirmed cyclic sieving for permutations with fixed cycle type and excedances

Developed new symmetric functions for analyzing permutation actions

Established a link between q-analogues and group actions in combinatorics

Abstract

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 $S_n$ 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.