International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 45-48, Pages 2509-2512
doi:10.1155/S0161171204311439
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
Abstract
Dyson defined the rank of a partition (as the first part minus the number of parts) whilst investigating certain congruences in the sequence p−1(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 p−2(n).