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arXiv · 2509.17541

Two-Dimensional Faces of Order and Chain Polytopes

Ragnar Freij-Hollanti·Teemu Lundström·Aki Mori
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Abstract

We give an explicit combinatorial description of the two-dimensional faces of both the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a partially ordered set $P$. Using these descriptions, we show that for any $P$, $\mathcal{C}(P)$ has equally many square faces, and at least as many triangular faces, as $\mathcal{O}(P)$ does. Moreover, the inequality is shown to be strict except when $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent. This proves the case $i=2$ of a conjecture by Hibi and Li.

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Ragnar Freij-Hollanti, Teemu Lundström, Aki Mori. 2025-09-22. Two-Dimensional Faces of Order and Chain Polytopes. https://arxiv.org/abs/2509.17541

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