2012/11/30 by Matthias Beck, Benjamin Braun, Matthias Köppe +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Context (archaeology) #Focus (optics) #Function (biology) #Integer (computer science) #Integer sequence #Mathematical functions and polynomials #Partition (number theory) #Real line #Sequence (biology) #math.AC #math.CO #math.NT #msc:05A17 #msc:05A19 #msc:13A02 #msc:13H10 #msc:52B11
paper · pdf · doi:10.1007/s11139-013-9538-3
published as Ramanujan Journal 36 (2015), 123-147
arxiv created 2013/10/17 · openalex publication_date 2014/01/13 · openalex created_date 2016/06/24 · arxiv updated 2017/01/03 · openalex updated_date 2026/08/05
In 1997, Bousquet-Melou 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-qe1)(1-qe2)...(1-qen))-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 greater than or equal to 1 if and only if s is an l-sequence. 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-qe1)(1-qe2)...(1-qen))-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.