Triangular faces of the order and chain polytope of a maximal ranked poset

Abstract

Let $O\left(P\right)$ and $C\left(P\right)$ denote the order polytope and chain polytope, respectively, associated with a finite poset P. We prove the following result: if P is a maximal ranked poset, then the number of triangular 2-faces of $O\left(P\right)$ is less than or equal to that of $C\left(P\right)$, with equality holding if and only if P does not contain an X-poset as a subposet.