Generalized snake posets, order polytopes, and lattice-point enumeration
Abstract
Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the Ehrhart polynomials of order polytopes of generalized snake posets along with a computation of their Gorenstein index. Then we give a combinatorial description of the chain polynomial of generalized snake posets as a direction to obtain the -polynomial of their associated order polytopes. Additionally, we present explicit formulae for the -polynomial of the order polytopes of the two extremal examples of generalized snake posets, namely the ladder and regular snake poset. We then provide a recursive formula for the -polynomial of any generalized snake posets and show that the -vectors are entry-wise bounded by the -vectors of the two extremal cases.
Cite
@article{arxiv.2411.18695,
title = {Generalized snake posets, order polytopes, and lattice-point enumeration},
author = {Eon Lee and Andrés R. Vindas-Meléndez and Zhi Wang},
journal= {arXiv preprint arXiv:2411.18695},
year = {2026}
}
Comments
30 pages, 9 figures, accepted for publication in Discrete Mathematics