Congruences in ordered pairs of partitions

1. Introducing the birank. A partition is defined as being a nonincreasing sequence of positive integers, λ = (λ1,λ2, . . . ,λr ). The set of all partitions, which includes the empty partition ∅, is denoted by . The sum of the parts of a given partition is called the weight of the partition, wt(λ) = λ1+λ2+···+λr . It is standard notation to write (z;q)∞ := ∏ t≥0(1−zq) and p−k(n) for the coefficient of qn in (q;q)−k ∞ , for fixed k. It is easy to show that the number of partitions of weight n is p−1(n) (for the empty partition, wt(λ)= 0). It is also easy to show that the number of ordered pairs of partitions of weight n is p−2(n), the weight of an ordered pair being defined as the sum of the weights of the two partitions in the pair. The sequence p−1(n) is known to satisfy certain congruences, one of which