  Boij-S\"oderberg theory shows that the Betti table of a graded module can be
written as a liner 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.
