The poset of proper divisibility

TL;DR

This paper investigates the poset of proper divisibility among monomials, proving shellability of its order complex, and provides explicit homology ranks and Euler characteristic formulas for the case of two variables.

Contribution

It establishes shellability of the order complex of the poset of proper divisibility and introduces a new example of a non-CL-shellable poset with a CL-shellable dual.

Findings

The order complex of the poset is shellable.

Explicit homology ranks are provided for the two-variable case.

A formula for the Euler characteristic of the order complex is derived.

Abstract

We study the partially ordered set $P(a_1,\ldots, a_n)$ 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 $P(a_1,\ldots a_n)$ is (non-pure) shellable, by showing that the order dual of $P(a_1,\ldots,a_n)$ 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 $\Delta \left( P(a_1,a_2) \right)$.