Level of regions for deformed braid arrangements

Abstract

This paper primarily investigates a specific type of deformation of the braid arrangement in $R^n$, denoted by $A_n^A$. Let $r_l\left(A_n^A\right)$ be the number of regions of level l in $A_n^A$ and $R_l(A;x)$ the corresponding exponential generating function. Using the weighted digraph model introduced by Hetyei, we establish a bijection between regions of level l in $A_n^A$ and valid m-acyclic weighted digraphs on the vertex set $\left[n\right]$ with exactly l strong components. Based on this bijection, we obtain that the sequence $R_1(A;x),R_2(A;x),\cdots$ is of binomial type. In addition, the values $r_l\left(A_n^A\right)$ provide a combinatorial interpretation for the coefficients when the characteristic polynomial of $A_n^A$ is expanded in terms of $\left(\begin{array}{c}t\\ l\end{array}\right)$. In particular, if $n\ge 2$ and $A=[-a,b]\cap Z$ for non-negative integers a and b with $b-a\ge n-1$, we show that the characteristic polynomial of $A_n^A$ 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.