The emergence of Boij–Söderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a -linear minimal resolution arises from that of the Stanley–Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of -linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with -linear resolutions, threshold graphs, and anti-lecture-hall compositions. Moreover, we prove that any Betti diagram of a module with a -linear resolution is realized by a direct sum of Stanley–Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.
Alexander Engström. Matthew Stamps. "Betti diagrams from graphs." Algebra Number Theory 7 (7) 1725 - 1742, 2013. https://doi.org/10.2140/ant.2013.7.1725