Counting regions of the boxed threshold arrangement

TL;DR

This paper introduces the boxed threshold arrangement in ^n, relates its bounded regions to labeled threshold graphs, and provides a bijective counting method with a closed-form formula for the total regions.

Contribution

It offers a new bijective approach to count regions of the arrangement, connecting them to signed partitions and colored threshold graphs, and derives a closed-form counting formula.

Findings

Number of regions equals the count of certain signed partitions and colored threshold graphs.

Established bijections between regions, signed partitions, and colored threshold graphs.

Derived a closed-form formula for the total number of regions.

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.