The Lecture Hall Parallelepiped

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.