# **s**-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones

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## Abstract

In 1997, Bousquet-Mélou and Eriksson initiated the study of *lecture hall partitions*, a fascinating family of partitions that yield a finite version of Euler’s celebrated odd/distinct partition theorem. In subsequent work on **s**-lecture hall partitions, they considered the *self-reciprocal property* for various associated generating functions, with the goal of characterizing those sequences **s** that give rise to generating functions of the form \(((1-q^{e_{1}})(1-q^{e_{2}}) \cdots(1-q^{e_{n}}))^{-1}\).

We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for **s**-lecture hall cones when **s** is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences **s**, we prove that the *n*-dimensional **s**-lecture hall cone is Gorenstein for all *n*≥1 if and only if **s** is an *ℓ*-sequence, i.e., recursively defined through *s* _{0}=0, *s* _{1}=1, and *s* _{ i }=*ℓs* _{ i−1}−*s* _{ i−2} for *i*≥2. One consequence is that among such sequences **s**, unless **s** is an *ℓ*-sequence, the generating function for the **s**-lecture hall partitions can have the form \(((1-q^{e_{1}})(1-q^{e_{2}}) \cdots(1-q^{e_{n}}))^{-1}\) for at most finitely many *n*.

We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of *h* ^{∗}-vectors for **s**-lecture hall polytopes. We end with open questions and directions for further research.

## Keywords

Lecture hall partition Polyhedral cone Generating function Gorenstein Self-reciprocal polynomial## Mathematics Subject Classification (2010)

05A17 05A19 52B11 13A02 13H10## Notes

### Acknowledgements

The authors thank the American Institute of Mathematics for support of our SQuaRE working group on “Polyhedral Geometry and Partition Theory.” We are deeply grateful to a referee who took time and care and did a thorough job of checking the paper. In the end (s)he understood the paper better than we did.

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