## 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.

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## References

- 1.
Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York (2007). Electronically available at http://math.sfsu.edu/beck/ccd.html

- 2.
Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. Ramanujan J.

**1**(1), 101–111 (1997) - 3.
Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. II. Ramanujan J.

**1**(2), 165–185 (1997) - 4.
Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)

- 5.
Ehrhart, E.: Sur les polyèdres rationnels homothétiques à

*n*dimensions. C. R. Acad. Sci. Paris**254**, 616–618 (1962) - 6.
Halava, V., Harju, T., Hirvensalo, M.: Positivity of second order linear recurrent sequences. Discrete Appl. Math.

**154**(3), 447–451 (2006) - 7.
Hochster, M.: Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes. Ann. Math. (2)

**96**, 318–337 (1972) - 8.
Loehr, N.A., Savage, C.D.: Generalizing the combinatorics of binomial coefficients via

*ℓ*-nomials. Integers**10**(A45), 531–558 (2010) - 9.
Pensyl, T.W., Savage, C.D.: Rational lecture hall polytopes and inflated Eulerian polynomials. Ramanujan J.

**31**, 97–114 (2013) - 10.
Savage, C.D., Schuster, M.J.: Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences. J. Comb. Theory, Ser. A

**119**, 850–870 (2012) - 11.
Savage, C.D., Viswanathan, G.: The 1/

*k*-Eulerian polynomials. Electron. J. Comb.**19**, P9 (2012). 21 pp. (electronic) - 12.
Savage, C.D., Yee, A.J.: Euler’s partition theorem and the combinatorics of

*ℓ*-sequences. J. Comb. Theory, Ser. A**115**(6), 967–996 (2008) - 13.
Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math.

**28**(1), 57–83 (1978)

## 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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## Additional information

*Dedicated to the memory of Basil Gordon*

Matthias Beck was partially supported by grant DMS-1162638 of the U.S. National Science Foundation. Benjamin Braun was partially supported by grant H98230-13-1-0240 of the U.S. National Security Agency. Matthias Köppe was partially supported by grant DMS-0914873 of the U.S. National Science Foundation. Carla Savage was partially supported by grant # 244963 from the Simons Foundation. Zafeirakis Zafeirakopoulos was supported by the strategic program “Innovatives OÖ 2010 plus” by the Upper Austrian Government and by the Austrian Science Fund (FWF) grants W1214-N15 (project DK6) and P22748-N18.

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### Cite this article

Beck, M., Braun, B., Köppe, M. *et al.*
**s**-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones.
*Ramanujan J* **36, **123–147 (2015). https://doi.org/10.1007/s11139-013-9538-3

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### Keywords

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

### Mathematics Subject Classification (2010)

- 05A17
- 05A19
- 52B11
- 13A02
- 13H10