Graded Expectations: Betti numbers and anti-lecture hall compositions of random threshold graphs

TL;DR

This paper explores the deep connections between threshold graphs, Betti numbers, and anti-lecture hall compositions, providing explicit combinatorial mappings and calculating expected values for random threshold graphs.

Contribution

It establishes new explicit combinatorial correspondences and computes expected Betti numbers and anti-lecture hall compositions for random threshold graphs.

Findings

Explicit combinatorial mappings between the objects

Calculated expected Betti numbers for random threshold graphs

Analyzed anti-lecture hall compositions in the context of threshold graphs

Abstract

This paper examines the one-to-one-to-one correspondence between threshold graphs, Betti numbers of quotients of polynomial rings by $2$-linear ideals, and anti-lecture hall compositions. In particular, we establish new explicit combinatorial mappings between each of these classes of objects and calculate the expected values of the Betti numbers and anti-lecture hall composition corresponding to a random threshold graph.