Generalized snake posets, order polytopes, and lattice-point enumeration

TL;DR

This paper explores the Ehrhart theory of order polytopes from generalized snake posets, providing formulas for their $h^*$-polynomials, and establishing bounds and properties of their Ehrhart and $h^*$-vectors.

Contribution

It introduces a recursive formula for the $h^*$-polynomial of generalized snake posets and characterizes their Ehrhart properties, extending previous work on order polytopes.

Findings

Explicit formulas for $h^*$-polynomials of ladder and snake posets

A recursive formula for $h^*$-polynomials of generalized snake posets

Bounds on $h^*$-vectors compared to extremal cases

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 $h^*$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^*$-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 $h^*$-polynomial of any generalized snake posets and show that the $h^*$-vectors are entry-wise bounded by the $h^*$-vectors of the two extremal cases.