Teemu Lundström, Leonardo Saud Maia Leite
Published 2025-04-07, updated 2025-05-05Version 2
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 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 $f$-vectors. 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).
Comments: 24 pages, 9 figures. Typos were fixed, Remark 4.3 was added, abstract was edited, and Corollary 5.19 was removed
Order-Chain Polytopes
The numbers of edges of the order polytope and the chain poyltope of a finite partially ordered set
The $h^*$-polynomial of the order polytope of the zig-zag poset
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