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… Expand

In this chapter, Beissinger and Peled proved that the number of labeled threshold graphs on $n\ge 2$ vertices is 2, and a direct combinatorial proof of this result is given.Expand

A multivariate generating function which counts signed permutations by their cycle type and to other descent statistics, analogous to a result of Gessel and Reutenauer for (unsigned) permutations is derived.Expand

This review of 3 Enumerative Combinatorics, by Charalambos A.good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations.Expand

A hyperplane arrangement is said to satisfy the “Riemann hypothesis” if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for… Expand