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

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.