International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 47, Pages 2509-2512
doi:10.1155/S0161171204311439
Congruences in ordered pairs of partitions
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.
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).