Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

Abstract

The Eulerian polynomial $\mathrm {AExc}_n(t)$ enumerating excedances in the symmetric group $\mathfrak {S}_n$ is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider $\mathrm {AExc}_n^+(t)$ and $\mathrm {AExc}_n^-(t)$, the polynomials which enumerate excedance in the alternating group $\mathcal {A}_n$ and in $\mathfrak {S}_n - \mathcal {A}_n$, respectively. We show that $\mathrm {AExc}_n^+(t)$ is gamma positive iff $n \ge 5$ is odd. When $n \ge 4$ is even, $\mathrm {AExc}_n^+(t)$ is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about $\mathrm {AExc}_n^-(t)$. We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in $\mathfrak {S}_n$. We extend some of these to the case when enumeration is done over even and odd derangements in $\mathfrak {S}_n$.