Abstract :

The Eulerian polynomial $A_n(t)$ is one of the most interesting and well-studied polynomials in combinatorics. The $k$-th coefficient of the $n$-th Eulerian polynomial counts the number of permutations in the symmetric group $\mathcal{S}_n$ with exactly k descents. The coefficient vectors of these polynomials satisfy all of combinatorists’ favorite distributional properties: palindromicity, gamma positivity, log-concavity etc. In this talk, we firstly consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathcal{S}_n \setminus \mathcal{A}_n$ respectively. We prove the gamma positivity of these polynomials, in part of the cases, depending on the values of $n$ mod $4$. We show similar gamma positivity results about the descent and excedance based type B and type D Eulerian polynomials when the enumeration is done over the positive elements in the respective Coxeter groups. Moving on, we will consider the alternating-runs enumerating polynomial over the symmetric group. There are three formulae for the number of permutations in $\mathcal{S}_n$ with $k$ alternating runs, but all of them are complicated. Here we will see that when enumerated with sign taken into account, one gets a neat formula. As a consequence of the signed enumeration, we get a near refinement of a result of Wilf on the exponent of $(1 + t)$ when it divides the alternating runs enumerating polynomial in the alternating group. Other applications include a moment-type identity and enumeration of alternating permutations in the alternating group. Time permitting, we will see log-concavity results and central limit theorems involving descents and excedances over positive elements of Coxeter groups.