Counting Labeled Threshold Graphs with Eulerian Numbers

Abstract

A threshold graph is any graph which can be constructed from the empty graph by repeatedly adding a new vertex that is either adjacent to every vertex or to no vertices. The Eulerian number n k counts the number of permutations of size n with exactly k ascents. Implicitly Beissinger and Peled proved that the number of labelled threshold graphs on n ≥ 2 vertices isTheir proof used generating functions. We give a direct combinatorial proof of this result.

Subset Parking Functions

“…Each term contributes 2 asc(π)−1 , so we conclude the result by Lemma 3.4 after noting that n−1 −1 = 0. Corollary 1.6 shows that, for n ≥ 2, IPF 2 (n) is equal to the number of connected threshold graphs on n vertices [3]. This can be proven bijectively from essentially the same proof as in [3], but for brevity we omit the details.…”

“…Corollary 1.6 shows that, for n ≥ 2, IPF 2 (n) is equal to the number of connected threshold graphs on n vertices [3]. This can be proven bijectively from essentially the same proof as in [3], but for brevity we omit the details. The formulas for k = 3 and k = n − 2 seem complicated (though the latter can be put into a closed form), and as of this writing neither sequence appears in the OEIS.…”

“…We note that this result is implicitly proven in [3], but for completeness we include the full proof here. To prove this, we recall the following recurrence for Eulerian numbers, which is valid for all n, k ≥ 1 [7].…”