The lecture hall parallelepiped (Q404540)

For a sequence \({\mathbf s}\) of positive integers, the \({\mathbf s}\)-lecture hall polytopes were defined by \textit{C. D. Savage} and \textit{M. J. Schuster} [J. Comb. Theory, Ser. A 119, No. 4, 850--870 (2012; Zbl 1237.05017)]. The lattice points in this polytope are in one-to-one correspondence with the \({\mathbf s}\)-lecture hall partitions, a class of partitions generalizing the lecture hall partitions defined by \textit{M. Bousquet-Mélou} and \textit{K. Eriksson} [Ramanujan J. 1, No. 1, 101--111 (1997; Zbl 0909.05008)]. This paper associates a half open parallelepiped to an \({\mathbf s}\)-lecture hall polytope and describes its lattice points. This description is used to recover earlier results of Savage et al. on connecting the generating function of the Ehrhart polynomials of \({\mathbf s}\)-lecture hall polytopes to \({\mathbf s}\)-ascents and \({\mathbf s}\)-descents, as well as some generalizations of these results.