Simplex inequalities of order and chain polytopes of recursively defined posets

TL;DR

This paper investigates the simplex faces of order and chain polytopes of recursively constructed posets, establishing inequalities in the number of simplex faces across different dimensions.

Contribution

It generalizes previous results by showing that for certain recursively built posets, chain polytopes have at least as many simplex faces as order polytopes in each dimension.

Findings

Chain polytopes have at least as many k-dimensional simplex faces as order polytopes.

The results extend to a broader class of posets constructed via disjoint unions and ordinal sums.

Generalizes Mori's previous findings on simplex faces of these polytopes.

Abstract

In this paper, we study the simplex faces of the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a finite poset $P$. We show that, if $P$ can be recursively constructed from $\mathbf{X}$-free posets using disjoint unions and ordinal sums, then $\mathcal{C}(P)$ has at least as many $k$-dimensional simplex faces as $\mathcal{O}(P)$ does, for each dimension $k$. This generalizes a previous result of Mori, both in terms of the dimensions of the simplices and in terms of the class of posets considered.