s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones

TL;DR

This paper explores the conditions under which s-lecture hall cones are Gorenstein, linking their properties to linear recurrence sequences and providing insights into the structure of associated generating functions and polytopes.

Contribution

It characterizes when s-lecture hall cones are Gorenstein for sequences generated by second-order recurrences, connecting this to l-sequences and analyzing generating function forms.

Findings

s-lecture hall cones are Gorenstein iff s is an l-sequence for all n

generating functions have specific forms only for finitely many n unless s is an l-sequence

established conjectures on symmetry of h*-vectors for s-lecture hall polytopes

Abstract

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research.