Generating functions and triangulations for lecture hall cones

We investigate the arithmetic-geometric structure of the lecture hall cone $$L_n:=\{\lambda \in {\mathbb{R}}^n:0\le \frac{{\lambda }_1}{1}\le \frac{{\lambda }_2}{2}\le \frac{{\lambda }_3}{3}\le \cdots \le \frac{{\lambda }_n}{n}\}\mathrm{.}$$ We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^{\ast }$-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 $L_n$ and cones over unit cubes.