The Primitive Eulerian polynomial

TL;DR

This paper introduces the Primitive Eulerian polynomial for hyperplane arrangements, providing a geometric and combinatorial interpretation of its coefficients for all simplicial arrangements, including types D and B, and explores its properties.

Contribution

It offers a new combinatorial and geometric interpretation of the Primitive Eulerian polynomial coefficients for all simplicial arrangements, extending understanding to type D and connecting to known Eulerian polynomials.

Findings

Coefficients have nonnegative values for simplicial arrangements.

Provides combinatorial interpretation for type D arrangements.

Connects Primitive Eulerian polynomial to the 1/2-Eulerian polynomial in type B.

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 $P_{\cal A}(z)$.