Gamma positivity of the descent based Eulerian polynomial in positive elements of classical Weyl groups

Hiranya Kishore Dey¹ ; Sivaramakrishnan Sivasubramanian²

¹Department of Mathematics, Indian Institute of Technology, Bombay, Mumbai 400 076, India.
²Dept of Mathematics,IIT Bombay

The electronic journal of combinatorics, Tome 27 (2020) no. 3

Résumé

The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ 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 known to be gamma positive for all $n$. We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively. We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$. We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.

Détail

DOI : 10.37236/9037

Classification : 05A05, 05A15, 20F55, 05E05
Mots-clés : Coxeter groups