The Lecture Hall Parallelepiped

TL;DR

This paper introduces a half-open parallelepiped associated with s-lecture hall polytopes, providing a new way to describe their integer points and recover known results related to their -vector, s-ascents, and s-descents.

Contribution

It defines a novel parallelepiped linked to s-lecture hall polytopes and offers a simplified description of its integer points, connecting to existing combinatorial properties.

Findings

Recovered earlier results on the -vector of P_s

Connected -vector to s-ascents and s-descents

Generalized previous combinatorial results

Abstract

The s-lecture hall polytopes P_s are a class of integer polytopes defined by Savage and Schuster which are closely related to the lecture hall partitions of Eriksson and Bousquet-M\'elou. We define a half-open parallelopiped Par_s associated with P_s and give a simple description of its integer points. We use this description to recover earlier results of Savage et al. on the \delta-vector (or h^*-vector) and to obtain the connections to s-ascents and s-descents, as well as some generalizations of these results.