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Simultaneous core partitions with nontrivial common divisor

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Abstract

A tremendous amount of research has been done in the last two decades on (s, t)-core partitions when s and t are relatively prime integers. Here we change perspective slightly and explore properties of (s, t)-core and \((\bar{s},\bar{t})\)-core partitions for s and t with a nontrivial common divisor g. We begin by recovering, using the g-core and g-quotient construction, the generating function for (s, t)-core partitions first obtained by Aukerman et al. (Discrete Math 309(9):2712–2720, 2009). Then, using a construction developed by the first two authors, we obtain a generating function for the number of \((\bar{s},\bar{t})\)-core partitions of n. Our approach allows for new results on t-cores and self-conjugate t-cores that are not g-cores and \(\bar{t}\)-cores that are not \(\bar{g}\)-cores, thus strengthening positivity results of Ono and Granville (Trans Am Soc 348:221–228, 1996), Baldwin et al. (J Algebra 297:438–452, 2006), and Kiming (J Number Theory 60:97–102, 1996). We then move to bijections between bar-core partitions and self-conjugate partitions. We give a new, short proof of a correspondence between self-conjugate t-core and \(\bar{t}\)-core partitions when t is odd and positive first due to Yang (Ramanujan J 44:197, 2019). Then, using two different lattice-path labelings, one due to Ford et al. (J Number Theory 129:858–865, 2009), the other to Bessenrodt and Olsson (J Algebra 306:3–16, 2006), we give a bijection between self-conjugate (s, t)-core and \((\bar{s},\bar{t})\)-core partitions when s and t are odd and coprime. We end this section with a bijection between self-conjugate (s, t)-core and \((\bar{s},\bar{t})\)-core partitions when s and t are odd and nontrivial g which uses the results stated above. We end the paper by noting (s, t)-core and \((\bar{s}, \bar{t})\)-core partitions inherit Ramanujan-type congruences from those of g-core and \(\bar{g}\)-core partitions.

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Acknowledgements

Part of this work was done at the Centre Interfacultaire Bernoulli (CIB), in the École Polytechnique Fédérale de Lausanne (Switzerland), during the Semester Local Representation Theory and Simple Groups. The first two authors are grateful to the CIB for their financial and logistical support. The first author also acknowledges financial support from the Engineering and Physical Sciences Research Council grant Combinatorial Representation Theory EP/M019292/1. The second author was supported by PSC-TRADA-46-493 and thanks George Andrews who supported a visit to Penn State where this research began. The second author also thanks Christopher R. H. Hanusa for helpful conversations on diagrams and references, and notes that some diagrams were made using the ytab package. All of the authors thank the anonymous referee for the careful reading and detailed and helpful suggestions.

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Authors and Affiliations

  1. Institute of Mathematics, University of Aberdeen, Kings College, Aberdeen, AB24 3UE, UK

    Jean-Baptiste Gramain

  2. Department of Mathematics, CUNY-York College, Jamaica, NY, 11451, USA

    Rishi Nath

  3. Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN, 55812, USA

    James A. Sellers

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  1. Jean-Baptiste Gramain
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  2. Rishi Nath
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  3. James A. Sellers
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Correspondence to Rishi Nath.

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Gramain, JB., Nath, R. & Sellers, J.A. Simultaneous core partitions with nontrivial common divisor. Ramanujan J 56, 839–863 (2021). https://doi.org/10.1007/s11139-020-00289-4

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