The Primitive Eulerian polynomial

TL;DR

Let $P_{ ext{A}}(z)$ denote the Primitive Eulerian polynomial of a finite real hyperplane arrangement $ ext{A}$, defined by $P_{ ext{A}}(z)= sum_{ ext{X} in mathcal{L}} | mu(ot, ext{X})| (z-1)^{ ext{codim}( ext{X})}$, and note its reparametrization of the cocharacteristic polynomial via $ Psi_{ ext{A}}(z)= sum | mu(ot, ext{X})| z^{ ext{dim}( ext{X})}$ with $P_{ ext{A}}(z)=(z-1)^{n} Psi_{ ext{A}}(1/(z-1))$. The paper demonstrates nonnegativity of coefficients for simplicial $ ext{A}$ through a geometric/combinatorial interpretation using generic halfspaces and the weak order, and provides explicit cusp/descent/excedance interpretations for reflection arrangements of types ${ m A}$, ${ m B}$, and ${ m D}$, including a new description for type ${ m D}$. It derives generating functions and recurrences for the Primitive Eulerian polynomials in types ${ m A}$, ${ m B}$, and ${ m D}$, and studies intermediate families between ${ m B}$ and ${ m D}$, such as $ ext{D}_{n,k}$. The work establishes real-rootedness for rank at most $3$ and for Coxeter/simplicial arrangements, while showing counterexamples in higher rank for non-simplicial cases and positing a broad conjecture on the real-rootedness of all simplicial Primitive Eulerian polynomials, clarifying connections to the $1/2$-Eulerian polynomials and related combinatorial statistics. This advances understanding of how hyperplane geometry encodes Eulerian-type statistics and opens avenues for applying these polynomials to toric and zonotopal combinatorics.

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

Figures (12)

• Figure 1: The Tits product for some faces of a rank 2 arrangement. After moving a small positive distance from the marked point in $F$ to the marked point in $G$, we land in the region labeled $FG$.
• Figure 2: The lattice of flats of the graphic arrangement of a 4-cycle. In red, the values of the Möbius function $\mu(\bot,\mathrm{X})$.
• Figure 3: Two copies of the spherical representation of an arrangement $\mathcal{A}$ in $\mathbb{R}^3$, each with a different hyperplane $\mathrm{H}$ not in $\mathcal{A}$ (dashed, in red). The hyperplane on the left is not generic since it contains the marked flat of rank $1$ of $\mathcal{A}$. The hyperplane on the right is generic.
• Figure 4: The arrangement and the bounding hyperplane of $\mathsf{h}$ are those in second example of \ref{['f:gen-hyp']}. In this picture we see the front and back view of $\mathcal{A}$. Observe that even though $\mathsf{h}$ is a convex set, the underlying set of $\Sigma(\mathsf{h})$ is not. We have that $\Psi_\mathcal{A}(z) = 1 + 7z + 12 z^2 + 6 z^3$ and $P_\mathcal{A}(z) = z^3 + 4 z^2 + z$.
• Figure 5: A non-simplicial arrangement $\mathcal{A}$. The faces $F$ (blue) and $G$ (red) have rank $1$ and $2$, respectively. The top-stars $\mathcal{R}_F$ and $\mathcal{R}_G$ (shaded) contain a minimum and maximum element in the order $\preceq_B$.

Theorems & Definitions (59)

• Theorem A
• Theorem B
• Definition 2.1
• Example 2.2: Rank 1 arrangement
• Remark 2.3
• Example 2.4: Rank 2 arrangements
• Proposition 2.5
• proof
• Example 2.6
• Proposition 2.7