Level of Regions for Deformed Braid Arrangements

Abstract

This paper primarily investigates a specific type of deformation of the braid arrangement $\mathcal{B}_n$ in $\mathbb{R}^n$, denoted by $\mathcal{B}_n^A$ and defined in (1.2). Let $r_l(\mathcal{B}_n^A)$ be the number of regions of level $l$ in $\mathcal{B}_n^A$ with the corresponding exponential generating function $R_l(A;x)$. Using the weighted digraph model introduced by Hetyei [11], we establish a bijection between regions of level $l$ in $\mathcal{B}_n^A$ and valid $m$-acyclic weighted digraphs on the vertex set $[n]$ with exactly $l$ strong components. Based on this bijection, we obtain a property analogous to a polynomial sequence of binomial type, that is, $R_l(A;x)$ satisfies the relation \[ R_l(A;x)=\big(R_1(A;x)\big)^l=R_k(A;x)R_{l-k}(A;x). \] Furthermore, the values $r_l(\mathcal{B}_n^A)$ yield a combinatorial interpretation for the coefficients in the expansion of the characteristic polynomial $\chi_{\mathcal{B}_n^A}(t)$ in the basis elements $\binom{t}{l}$, that is, \[\chi_{\mathcal{B}_n^A}(t)=\sum_{l=0}^n(-1)^{n-l}r_l(\mathcal{B}_n^A)\binom{t}{l}.\] If $n$, $a$ and $b$ are non-negative integers with $n\ge 2$ and $b-a\ge n-1$, for the deformation $\mathcal{B}_n^{[-a,b]}$ defined in (1.3), its characteristic polynomial has a single real root $0$ of multiplicity one when $n$ is odd, and has one more real root $\frac{n(a+b+1)}{2}$ of multiplicity one when $n$ is even.