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ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

  • MICHAEL D. HIRSCHHORN (a1) and JAMES A. SELLERS (a2)
Abstract
Abstract

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

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Corresponding author
For correspondence; e-mail: sellersj@math.psu.edu
References
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[1] Baruah N. and Berndt B., ‘Partition identities and Ramanujans modular equations’, J. Combin. Theory Ser. A 114(6) (2007), 10241045.
[2] Borwein J. and Borwein P., ‘A cubic counterpart of Jacobi’s identity and the AGM’, Trans. Amer. Math. Soc. 323(2) (1991), 691701.
[3] Borwein J., Borwein P. and Garvan F., ‘Some cubic modular identities of Ramanujan’, Trans. Amer. Math. Soc. 343(1) (1994), 3547.
[4] Granville A. and Ono K., ‘Defect zero p-blocks for finite simple groups’, Trans. Amer. Math. Soc. 348(1) (1996), 331347.
[5] Hirschhorn M., ‘A Letter from Fitzroy House’, Amer. Math. Monthly 115(6) (2008), 563566.
[6] Hirschhorn M., Garvan F. and Borwein J., ‘Cubic analogues of the Jacobian theta function θ(z,q)’, Canad. J. Math. 45(4) (1993), 673694.
[7] James G. and Kerber A., The Representation Theory of the Symmetric Group (Addison–Wesley Publishing, Reading, MA, 1981).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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