#### 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.

Done