Chandrasekhar Hall

Gamma positivity of the Exceedance Based Eulerian Polynomial in positive elements of Classical Weyl Groups

Sivaramakrishnan Sivasubramanian

IIT Bombay

The Eulerian polynomial $\GE_n(t)$ enumerating excedances in
$S_n$ 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 the polynomials $\GE_n^+(t)$ and $\GE_n^-(t)$ which enumerate
excedance in the alternating group $A_n$ and in $S_n - A_n$ respectively. We
show that $GE_n^+(t)$ is gamma positive iff $n \geq 5$ is odd.
When $n \geq 4$ is even, $\GE_n^+(t)$ is not even palindromic,
but we show that it is the sum of two gamma positive summands.
An identical statement is true about $\GE_n^-(t)$.

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.

This is joint work with Hiranya Dey.