A combinatorial statistic for labeled threshold graphs

TL;DR

This paper establishes a combinatorial interpretation for the coefficients of the characteristic polynomial of the threshold arrangement, linking them to labeled threshold graphs with specific properties, thus answering a question by Stanley.

Contribution

It provides a novel combinatorial meaning for the polynomial coefficients of the threshold arrangement, connecting hyperplane arrangements with graph properties.

Findings

Coefficients correspond to counts of labeled threshold graphs with certain properties

The work answers a question posed by Stanley about the combinatorial interpretation of these coefficients

Establishes a link between hyperplane arrangements and graph theory

Abstract

Consider the collection of hyperplanes in $\mathbb{R}^n$ whose defining equations are given by $\{x_i + x_j = 0\mid 1\leq i<j\leq n\}$. This arrangement is called the threshold arrangement since its regions are in bijection with labeled threshold graphs on $n$ vertices. Zaslavsky's theorem implies that the number of regions of this arrangement is the sum of coefficients of the characteristic polynomial of the arrangement. In the present article we give a combinatorial meaning to these coefficients as the number of labeled threshold graphs with a certain property, thus answering a question posed by Stanley.