Abstract
In the interest of finding the minimum additive generating set for the set of \({\varvec{s}}\)-lecture hall partitions, we compute the Hilbert bases for the \({\varvec{s}}\)-lecture hall cones in certain cases. In particular, we determine the Hilbert bases for two well-studied families of sequences, namely the \(1\mod k\) sequences and the \(\ell \)-sequences. Additionally, we provide a characterization of the Hilbert bases for \({\varvec{u}}\)-generated Gorenstein \({\varvec{s}}\)-lecture hall cones in low dimensions.
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Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York (2007)
Beck, M., Braun, B., Köppe, M., Savage, C.D., Zafeirakopoulos, Z.: s-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones. Ramanujan J. 36(1–2), 123–147 (2015)
Beck, M., Braun, B., Köppe, M., Savage, C.D., Zafeirakopoulos, Z.: Generating functions and triangulations for lecture hall cones. SIAM J. Discret. Math. 30(3), 1470–1479 (2016)
Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. Ramanujan J. 1(1), 101–111 (1997)
Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. II. Ramanujan J. 1(2), 165–185 (1997)
Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
Bruns, W., Ichim, B., Römer, T., Sieg, R., Söger, C.: Normaliz. Algorithms for rational cones and affine monoids. https://www.normaliz.uni-osnabrueck.de
Ehrhart, E.: Sur les polyèdres homothétiques bordés à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 988–990 (1962)
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics. Springer, New York (2005)
Pensyl, T.W., Savage, C.D.: Lecture hall partitions and the wreath products \(C_k\wr S_n\). Integers. In: Proceedings of the Integers Conference 2011, vol. 12B, Paper No. A10, 18 (2012/13)
Pensyl, T.W., Savage, C.D.: Rational lecture hall polytopes and inflated Eulerian polynomials. Ramanujan J. 31(1–2), 97–114 (2013)
Savage, C.D.: The mathematics of lecture hall partitions. J. Comb. Theory Ser. A 144, 443–475 (2016)
Savage, C.D., Viswanathan, G.: The \(1/k\)-Eulerian polynomials. Electron. J. Comb. 19(1), P9 (2012)
Savage, C.D., Yee, A.J.: Euler’s partition theorem and the combinatorics of \(\ell \)-sequences. J. Comb. Theory Ser. A 115(6), 967–996 (2008)
Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)
Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, 2nd edn. Birkhäuser Boston, Inc., Boston, MA (1996)
Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49, 2nd edn. Cambridge University Press, Cambridge (2012)
Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence, RI (1996)
Acknowledgements
The author thanks the American Institute of Mathematics, as this work began at the November 2016 workshop on polyhedral geometry and partition theory. The author thanks his advisor, Benjamin Braun, for helpful comments and suggestions throughout this project. The author also thanks the anonymous referees for reading the manuscript carefully and providing helpful suggestions and comments.
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Olsen, M. Hilbert bases and lecture hall partitions. Ramanujan J 47, 509–531 (2018). https://doi.org/10.1007/s11139-017-9933-2
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DOI: https://doi.org/10.1007/s11139-017-9933-2