This talk will explore a connection between seemingly unrelated families of graphs, integer sequences, and numerical invariants of) polynomial rings, specifically threshold graphs, anti-lecture hall compositions, and Betti numbers of quotients of polynomial rings with 2-linear resolutions. I will introduce these three families of objects, describe an explicit one-to-one-to-one correspondence between them, and elucidate some preliminary consequences of said correspondence. Most notably, I will make use of a well-studied random model for threshold graphs to calculate expected values (and properties) of Betti numbers and anti-lecture hall compositions. No prior knowledge of commutative algebra will be assumed; all of the essential ideas will be illustrated with concrete examples. This is joint work with Alexander Engström and Christian Go.