On the joint distribution of cyclic valleys and excedances over conjugacy classes of $\mathfrak{S}_{n}$

Abstract

We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's cyclic valley-hopping action as well as a formula of Brenti. Along the way, we give a new proof for the $\gamma$-positivity of the excedance number distribution over any fixed conjugacy class along with a combinatorial interpretation of the $\gamma$-coefficients.