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The Ramanujan Journal

, Volume 36, Issue 1–2, pp 123–147 | Cite as

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

  • Matthias Beck
  • Benjamin BraunEmail author
  • Matthias Köppe
  • Carla D. Savage
  • Zafeirakis Zafeirakopoulos
Article

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 -sequence, i.e., recursively defined through s 0=0, s 1=1, and s i =ℓs i−1s i−2 for i≥2. One consequence is that among such sequences s, unless s is an -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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Matthias Beck
    • 1
  • Benjamin Braun
    • 2
    Email author
  • Matthias Köppe
    • 3
  • Carla D. Savage
    • 4
  • Zafeirakis Zafeirakopoulos
    • 5
  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA
  3. 3.Department of MathematicsUniversity of California, DavisDavisUSA
  4. 4.Department of Computer ScienceNorth Carolina State UniversityRaleighUSA
  5. 5.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria

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