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On recent congruence results of Andrews and Paule for broken k-diamonds

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, NSW 2052, Australia, e-mail: m.hirschhorn@unsw.edu.au
James A. Sellers
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, e-mail: sellersj@math.psu.edu
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In one of their most recent works, George Andrews and Peter Paule continue their study of partition functions via MacMahon's Partition Analysis by considering partition functions associated with directed graphs which consist of chains of hexagons. In the process, they prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. Our first goal in this note is to reprove this congruence by explicitly finding the generating function in question. We then prove one of the conjectures posed by Andrews and Paule as well as a number of congruences not mentioned by them. All of our results follow from straightforward generating function manipulations.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 1 , February 2007 , pp. 121 - 126
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Andrews, G.E. and Paule, P., ‘MacMahon's partition analysis XI: Hexagonal plane partitions’, Acta Arith. (to appear).Google Scholar
[2]Hirschhorn, M.D., Garvan, F. and Borwein, J., ‘Cubic analogues of the Jacobian theta function θ(z, q)’ Canad. J. Math. 45 (1993), 673–694.CrossRefGoogle Scholar
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[4]MacMahon, P.A., Combinatory analysis, (2 vols.) (Cambridge University Press, Cambridge-1916). (Reprinted: Chelsea, New York, 1960).Google Scholar