Level of Regions for Deformed Braid Arrangements

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 $$R_l(A;x)=(R_1(A;x){)}^l=R_k(A;x)R_{l-k}(A;x)\mathrm{.}$$ 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 $\left(\genfrac{}{}{0px}{}{t}{l}\right)$, that is, $${\chi }_{{\mathcal{B}}_n^A}(t)=\sum _{l=0}^n(-1{)}^{n-l}{r}_l({\mathcal{B}}_n^A)\left(\genfrac{}{}{0px}{}{t}{l}\right)\mathrm{.}$$ 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.