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 ).
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Hammond et al. (2004) studied this question.
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