The Primitive Eulerian polynomial

Abstract

We introduce the Primitive Eulerian polynomial $P_{\cal A}(z)$ of a central hyperplane arrangement ${\cal A}$. It is a reparametrization of its cocharacteristic polynomial. Previous work of the first author implicitly show that, for simplicial arrangements, $P_{\cal A}(z)$ has nonnegative coefficients. For reflection arrangements of type A and B, the same work interprets the coefficients of $P_{\cal A}(z)$ using the (flag)excedance statistic on (signed) permutations. The main result of this article is to provide an interpretation of the coefficients of $P_{\cal A}(z)$ for all simplicial arrangements only using the geometry and combinatorics of ${\cal A}$. This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial meaning to the coefficients of the Primitive Eulerian polynomial of the reflection arrangement of type D. In type B, we find a connection between the Primitive Eulerian polynomial and the $1/2$-Eulerian polynomial of Savage and Viswanathan (2012). We present some real-rootedness results and conjectures for $P_{\cal A}(z)$.