The poset of proper divisibility

Abstract

We study the partially ordered set P(a1, ... , an) of all multidegrees (b1, ... , bn) of monomials xb1 1 ... xbn n, which properly divide xa1 1 ... xan n . We prove that the order complex Δ(P(a1, ... , an)) of P(a1, ... an) is (nonpure) shellable by showing that the order dual of P(a1, ... , an) is CL-shellable. Along the way, we exhibit the poset P(4, 4) as a new example of a poset with CL-shellable order dual that is not CL-shellable itself. For n = 2, we provide the rank of all homology groups of the order complex δ(P(a1, a2)). Furthermore, we give a succinct formula for the Euler characteristic of δ(P(a1, a2)).