Hilbert Bases and Lecture Hall Partitions

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