Counting regions of the boxed threshold arrangement

Abstract: In this paper we consider the hyperplane arrangement in $\mathbb{R}^n$ whose hyperplanes are ${x_i + x_j = 1\mid 1\leq i < j\leq n}\cup {x_i=0,1\mid 1\leq i\leq n}$. We call it the \emph{boxed threshold arrangement} since we show that the bounded regions of this arrangement are contained in an $n$-cube and are in one-to-one correspondence with the labeled threshold graphs on $n$ vertices. The problem of counting regions of this arrangement was studied earlier by Joungmin Song. He determined the characteristic polynomial of this arrangement by relating its coefficients to the count of certain graphs. Here, we provide bijective arguments to determine the number of regions. In particular, we construct certain signed partitions of the set ${-n,\dots, n}\setminus{0}$ and also construct colored threshold graphs on $n$ vertices and show that both these objects are in bijection with the regions of the boxed threshold arrangement. We independently count these objects and provide closed form formula for the number of regions.