International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 47, Pages 2509-2512

Congruences in ordered pairs of partitions

Paul Hammond1 and Richard Lewis2

1Department of mathematics, University of Sussex, Falmer, Brighton BN1 9RF, UK
2Department of Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK

Received 28 November 2003; Revised 12 December 2003

Copyright © 2004 Paul Hammond and Richard Lewis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Dyson defined the rank of a partition (as the first part minus the number of parts) whilst investigating certain congruences in the sequence p1(n). The rank has been widely studied as have been other statistics, such as the crank. In this paper a “birank” is defined which relates to ordered pairs of partitions, and is used in an elementary proof of a congruence in p2(n).