Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices

TL;DR

This paper investigates Ehrhart positivity and unimodular triangulations of s-lecture hall simplices, providing new counterexamples to positivity and explicit constructions of triangulations for certain classes.

Contribution

It introduces a new class of sequences s where s-lecture hall simplices are not Ehrhart positive and extends known classes with explicit unimodular triangulations.

Findings

Identified sequences s with negative Ehrhart coefficients.

Constructed explicit unimodular triangulations for certain classes.

Extended classes of s-lecture hall simplices with known triangulation properties.

Abstract

Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit.