Generating functions and triangulations for lecture hall cones

We investigate the arithmetic-geometric structure of the lecture hall cone L_n := {∈ ℝⁿ : 0 ≤ 1/1 ≤ 2/1 ≤ 3/1 ≤ ...≤ n/n}. We show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h^∗-polynomial is given by the (n-1)st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for L_n, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of L_n, including connections between enumerative and algebraic properties of Ln and cones over unit cubes.