ON THE NUMBER OF EVEN AND ODD STRINGS ALONG THE OVERPARTITIONS OF n

A partition of a non-negative integer n is a non-increasing sequence of positive integers whose sum is n. For example, The 5 partitions of 4 are 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. To explain Ramanujan’s famous three partition congruences, the partition statistics, the rank (resp. the crank) were introduced by Dyson [8] (resp. Andrews and Garvan [3]). Atkin and Garvan [5] initiated the study of rank and crank moments to examine the relations between the rank and the crank. For k ≥ 1, the k-th rank moment Nk(n) and the k-th crank moment Mk(n) are given by