Combinatorial Theory

We introduce the primitive Eulerian polynomial \(P_{\mathcal{A}}(z)\) of a central hyperplane arrangement \(\mathcal{A}\). It is a reparametrization of its cocharacteristic polynomial. Previous work by the first author implicitly shows that for simplicial arrangements, \(P_{\mathcal{A}}(z)\) has nonnegative coefficients. For reflection arrangements of types A and B, the same work interprets the coefficients of \(P_{\mathcal{A}}(z)\) using the (flag) excedance statistic on (signed) permutations.

The main result of this article is to provide an interpretation of the coefficients of \(P_{\mathcal{A}}(z)\) for all simplicial arrangements using only the geometry and combinatorics of \(\mathcal{A}\). This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial significance to the coefficients of the primitive Eulerian polynomial of the reflection arrangement of type D, for which no well-behaved excedance statistic is known. In type B, we establish a link between the primitive Eulerian polynomial and the \(1/2\)-Eulerian polynomial of Savage and Viswanathan (2012). We present some results and conjectures regarding the real-rootedness of \(P_{\mathcal{A}}(z)\).

Mathematics Subject Classifications: 52C35, 05A05

Keywords: Hyperplane arrangement, Eulerian polynomial, Tits product, permutation statistics