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arXiv · 1508.04619

Generating functions and triangulations for lecture hall cones

Matthias Beck·Benjamin Braun·Matthias Köppe·Carla D. Savage·Zafeirakis Zafeirakopoulos
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Abstract

We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq \frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq \frac{\lambda_n}{n}\right\} . \] We show that $L_n$ 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 $L_n$ and cones over unit cubes.

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Matthias Beck, Benjamin Braun, Matthias Köppe, Carla D. Savage, Zafeirakis Zafeirakopoulos. 2015-08-19. Generating functions and triangulations for lecture hall cones. https://doi.org/10.1137/15m1036907

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