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Numerical Sets, Core Partitions, and Integer Points in Polytopes

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

Abstract

We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results about core partitions. For small values of a, we give formulas for the number of (ab)-core partitions corresponding to numerical semigroups. We also study the number of partitions with a given hook set.

Keywords

  • Numerical semigroups
  • Numerical sets
  • Core partitions
  • Simultaneous core partitions
  • Hook sets of partitions

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Acknowledgements

The third author thanks Mel Nathanson for organizing the 2010 CANT conference where he first learned about core partitions and their connection to numerical semigroups in a talk by William Keith. He also thanks Maria Monks Gillespie for helpful discussions on early parts of this project.

We would like to thank Florencia Orosz-Hunziker and Daniel Corey for their assistance throughout this project. Finally, we would like to thank the Summer Undergraduate Math Research at Yale program for organizing, funding, and supporting this project. SUMRY is supported in part by NSF grant CAREER DMS-1149054.

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Constantin, H., Houston-Edwards, B., Kaplan, N. (2017). Numerical Sets, Core Partitions, and Integer Points in Polytopes. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_7

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