# Hilbert Bases and Lecture Hall Partitions

## TL;DR

This paper computes Hilbert bases for specific families of lecture hall cones to understand their minimal additive generators, providing new insights into their algebraic structure.

## Contribution

It introduces methods to compute Hilbert bases for certain lecture hall cones, including well-studied sequences and low-dimensional Gorenstein cases.

## Key findings

- Hilbert bases computed for $1 ext{ mod }k$ sequences
- Hilbert bases computed for $ ext{l}$-sequences
- Characterization of Hilbert bases for low-dimensional Gorenstein cones

## Abstract

In the interest of finding the minimum additive generating set for the set of $\boldsymbol{s}$-lecture hall partitions, we compute the Hilbert bases for the $\boldsymbol{s}$-lecture hall cones in certain cases. In particular, we compute the Hilbert bases for two well-studied families of sequences, namely the $1\mod k$ sequences and the $\ell$-sequences. Additionally, we provide a characterization of the Hilbert bases for $\boldsymbol{u}$-generated Gorenstein $\boldsymbol{s}$-lecture hall cones in low dimensions.