The poset of proper divisibility

Abstract

We study the partially ordered set $P(a_1,\dots ,a_n)$ of all multidegrees $(b_1,\dots ,b_n)$ of monomials $x_1^{b_1}\cdots x_n^{b_n}$ which properly divide $x_1^{a_1}\cdots x_n^{a_n}$. We prove that the order complex $\Delta (P(a_1,\dots ,a_n))$ of $P(a_1,\dots a_n)$ is (non-pure) shellable, by showing that the order dual of $P(a_1,\dots ,a_n)$ is $\mathrm{CL}$-shellable. Along the way, we exhibit the poset $P(4,4)$ as a new example of a poset with $\mathrm{CL}$-shellable order dual that is not $\mathrm{CL}$-shellable itself. For $n=2$ we provide the rank of all homology groups of the order complex $\Delta \left(P(a_1,a_2)\right)$. Furthermore, we give a succinct formula for the Euler characteristic of $\Delta \left(P(a_1,a_2)\right)$.