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The Ramanujan Journal

, Volume 47, Issue 3, pp 509–531 | Cite as

Hilbert bases and lecture hall partitions

  • McCabe Olsen
Article
  • 99 Downloads

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.

Keywords

Lecture hall partitions Hilbert bases Gorenstein cones 

Mathematics Subject Classification

05A17 05A19 11P21 13A02 13H10 13P99 52B11 

Notes

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.

References

  1. 1.
    Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. Ramanujan J. 1(1), 101–111 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. II. Ramanujan J. 1(2), 165–185 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  7. 7.
    Bruns, W., Ichim, B., Römer, T., Sieg, R., Söger, C.: Normaliz. Algorithms for rational cones and affine monoids. https://www.normaliz.uni-osnabrueck.de
  8. 8.
    Ehrhart, E.: Sur les polyèdres homothétiques bordés à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 988–990 (1962)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics. Springer, New York (2005)Google Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Pensyl, T.W., Savage, C.D.: Rational lecture hall polytopes and inflated Eulerian polynomials. Ramanujan J. 31(1–2), 97–114 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Savage, C.D.: The mathematics of lecture hall partitions. J. Comb. Theory Ser. A 144, 443–475 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Savage, C.D., Viswanathan, G.: The \(1/k\)-Eulerian polynomials. Electron. J. Comb. 19(1), P9 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, 2nd edn. Birkhäuser Boston, Inc., Boston, MA (1996)zbMATHGoogle Scholar
  17. 17.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49, 2nd edn. Cambridge University Press, Cambridge (2012)Google Scholar
  18. 18.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence, RI (1996)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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