Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-27T10:41:32.090Z Has data issue: false hasContentIssue false

ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

Published online by Cambridge University Press:  17 April 2009

MICHAEL D. HIRSCHHORN
Affiliation:
School of Mathematics and Statistics, UNSW, Sydney 2052, Australia (email: m.hirschhorn@unsw.edu.au)
JAMES A. SELLERS*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (email: sellersj@math.psu.edu)
*
For correspondence; e-mail: sellersj@math.psu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Baruah, N. and Berndt, B., ‘Partition identities and Ramanujans modular equations’, J. Combin. Theory Ser. A 114(6) (2007), 10241045.CrossRefGoogle Scholar
[2] Borwein, J. and Borwein, P., ‘A cubic counterpart of Jacobi’s identity and the AGM’, Trans. Amer. Math. Soc. 323(2) (1991), 691701.Google Scholar
[3] Borwein, J., Borwein, P. and Garvan, F., ‘Some cubic modular identities of Ramanujan’, Trans. Amer. Math. Soc. 343(1) (1994), 3547.CrossRefGoogle Scholar
[4] Granville, A. and Ono, K., ‘Defect zero p-blocks for finite simple groups’, Trans. Amer. Math. Soc. 348(1) (1996), 331347.CrossRefGoogle Scholar
[5] Hirschhorn, M., ‘A Letter from Fitzroy House’, Amer. Math. Monthly 115(6) (2008), 563566.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[7] James, G. and Kerber, A., The Representation Theory of the Symmetric Group (Addison–Wesley Publishing, Reading, MA, 1981).Google Scholar