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arXiv · 2504.05123

Order polytopes of crown posets

Teemu Lundström·Leonardo Saud Maia Leite
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Abstract

In the last decade, the order polytope of the zigzag poset has been thoroughly studied. A related poset, called \emph{crown poset}, obtained by adding an extra cover relation between the endpoints of an even zigzag poset, is not so well understood. In this paper, we study the order polytopes of crown posets. We provide explicit formulas for their $f$-polynomials, and show that the sequence of coefficients of each $f$-polynomial is log-concave and asymptotically normal. We provide recursive formulas for their Ehrhart polynomial, giving a counterpart to formulas found in the zigzag case by Petersen--Zhuang (2025). We use these formulas to simplify a computation by Ferroni--Morales--Panova (2025) of the linear term of the order polynomial of these posets. Furthermore, we provide a combinatorial interpretation for the coefficients of the $h^*$-polynomial in terms of the cyclic swap statistic on cyclically alternating permutations, which provides a circular version of a result by Coons--Sullivant (2023).

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Teemu Lundström, Leonardo Saud Maia Leite. 2026-09-22. Order polytopes of crown posets. https://doi.org/10.1016/j.ejc.2025.104304

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