## 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 func-tions, with the goal of characterizing those sequences s that give rise to generating functions of the form ((1 − q^{e1})(1 − q^{e2}) · · · (1 −q^{en})) ^{−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 s0 = 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^{e1})(1 −q^{e2}) · · · (1 −q^{en})) −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.

Original language | English |
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Pages (from-to) | 123-147 |

Number of pages | 25 |

Journal | Ramanujan Journal |

Volume | 36 |

Issue number | 1 |

DOIs | |

State | Published - Jan 14 2014 |

### Bibliographical note

Publisher Copyright:© Springer Science+Business Media New York 2014.

## Keywords

- Generating function
- Gorenstein
- Lecture hall partition
- Polyhedral cone
- Self-reciprocal polynomial

## ASJC Scopus subject areas

- Algebra and Number Theory