A combinatorial statistic for labeled threshold graphs

Abstract

Consider the collection of hyperplanes in R whose defining equations are given by {xi + xj = 0 | 1 ≤ i < j ≤ 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.