The poset of proper divisibility

We study the partially ordered set P(a₁, ... , a_n) of all multidegrees (b₁, ... , b_n) of monomials x_{b1 1} ... x^bn _n, which properly divide x_a1 1 ... x^an _n . We prove that the order complex Δ(P(a₁, ... , a_n)) of P(a₁, ... an) is (nonpure) shellable by showing that the order dual of P(a₁, ... , a_n) 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)).