Polyhedral geometry for lecture hall partitions

TL;DR

This paper reviews recent advances in the polyhedral geometric aspects of lecture hall partitions, emphasizing new results on lecture hall cones, simplices, and order polytopes, and discusses open problems in the field.

Contribution

It provides a focused survey on recent polyhedral geometric developments in lecture hall partitions, highlighting new results and open problems since prior comprehensive surveys.

Findings

New results on lecture hall cones and simplices

Advances in Ehrhart theory related to lecture hall partitions

Identification of open problems and conjectures in the area

Abstract

Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial results, which are of interest in combinatorial number theory, enumerative combinatorics, and convex geometry. In a recent survey of Savage \cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented. However, since the publication of this survey, there have been many new developments related to the polyhedral geometry and Ehrhart theory arising from lecture hall partitions. Subsequently, in this survey article, we focus exclusively on the polyhedral geometric results in the theory of lecture hall partitions in an effort to showcase these new developments. In particular, we highlight results on lecture hall cones, lecture hall simplices, and lecture hall order polytopes. We conclude with an extensive list of open problems and conjectures in this area.