S-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones

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, s₁ = 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.