Jose Bastidas, LACIM
Title: The primitive Eulerian polynomial.
Abstract: We introduce the Primitive Eulerian polynomial $P_mathcal{A}(z)$ of a central hyperplane Arrangement $mathcal{A}$. It is a reparametrization of the cocharacteristic polynomial of the arrangement. Previous work (2021) implicitly showed that this polynomial has nonnegative coefficients in the simplicial case. If $mathcal{A}$ is the arrangement corresponding to a Coxeter group $W$ of type A or B, then $P_mathcal{A}(z)$ is the generating function for the (flag)excedance statistic on a particular subset of $W$. No interpretation was found for reflection arrangements of type D.
We present an alternative geometric and combinatorial interpretation for the coefficients of $P_mathcal{A}(z)$ for all simplicial arrangements $mathcal{A}$. For reflection arrangements of types A, B, and D, we find recursive formulas that mirror those for the Eulerian polynomial of the corresponding type. We also present real-rootedness results and conjectures for $P_mathcal{A}(z)$. This is joint work with Christophe Hohlweg and Franco Saliola.