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

TL;DR

This paper derives a formula for the joint distribution of cyclic valleys and excedances over conjugacy classes in symmetric groups, linking it to Eulerian polynomials and providing new combinatorial insights.

Contribution

It extends cyclic valley-hopping actions and Brenti's formula to analyze joint distributions, also proving $b3$-positivity with combinatorial interpretation.

Findings

Joint distribution formula expressed via Eulerian polynomials

New proof of $b3$-positivity for excedance distribution

Combinatorial interpretation of $b3$-coefficients

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.