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Counting partitions on the abacus

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Abstract

In 2003, Maróti showed that one could use the machinery of ℓ-cores and ℓ-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case ℓ=2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss.

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References

  1. Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatorial analysis. Proc. London Math. Soc. (2) 17, 75–113 (1917)

    Article  Google Scholar 

  2. Erdős, P.: On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. (2) 43, 437–450 (1942)

    Article  Google Scholar 

  3. Gauss, C.F.: Zur Theorie der Transcendenten Functionen Gehörig. In: Werke, vol. III, pp. 435–446. Königslichen Gesellschaft der Wissenschaften zu Göttingen, Göttingen (1886)

    Google Scholar 

  4. Newman, D.J.: The evaluation of the constant in the formula for the number of partitions of n. Am. J. Math. 73, 599–601 (1951)

    Article  Google Scholar 

  5. James, G., Kerber, A.: The Representation Theory of The Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16. Addison-Wesley, Reading (1981)

    MATH  Google Scholar 

  6. Andrews, G.E.: Two theorems of Gauss and allied identities proved arithmetically. Pac. J. Math. 41, 563–578 (1972)

    MATH  Google Scholar 

  7. Pak, I.: Partition bijections, a survey. Ramanujan J. 12, 5–75 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Maróti, A.: On elementary lower bounds for the partition function. Integers 3, A10 (2003) (electronic, 9 pp.)

    Google Scholar 

  9. Külshammer, B., Olsson, J.B., Robinson, G.R.: Generalized blocks for symmetric groups. Invent. Math. 151(3), 513–552 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departament of Mathematics, University of Wales, Swansea, Singleton Park, Swansea, SA2 8PP, UK

    Mark Wildon

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  1. Mark Wildon
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Correspondence to Mark Wildon.

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Dedicated to the memory of Dr. Manfred Schocker (1970–2006)

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Wildon, M. Counting partitions on the abacus. Ramanujan J 17, 355–367 (2008). https://doi.org/10.1007/s11139-006-9013-5

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  • DOI: https://doi.org/10.1007/s11139-006-9013-5

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