Combinatorial interpretations of some Boij–Söderberg decompositions

Boij–Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.