  Boij and S\"oderberg made a pair of conjectures, which were subsequently
proven by Eisenbud and Schreyer and then extended by Boij and S\"oderberg,
about the structure of Betti diagrams of Graded modules. In the theory, a
particular family of posets, and their associated order complexes, play an
integral role. We explore the structure of this family. In particular, we show
the posets are bounded complete lattices and the order complexes are
vertex-decomposable, hence Cohen-Macaulay and squarefree glicci.

Combinatorics