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ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  17 April 2009

MICHAEL D. HIRSCHHORN  and
JAMES A. SELLERS
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
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Abstract

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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).

Keywords

partition3-coresgenerating functionJacobi’s Triple Product IdentitycongruencesLambert series

MSC classification

Secondary: 05A17: Partitions of integers 11P81: Elementary theory of partitions 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 3 , June 2009 , pp. 507 - 512
Copyright
Copyright © Australian Mathematical Society 2009

References

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