The odd moments of ranks and cranks

have motivated much research. Here, p(n) denotes the number of partitions of n. In particular, toward a combinatorial explanation of the above congruences many partition statistics have been studied. Among them, the rank suggested by F. Dyson [6] and the crank suggested by the first author and F.G. Garvan [2] have proven successful and their own properties have been extensively studied. Here, the rank of partition λ is defined by λ1 − `(λ), where λ1 is the largest part of λ and `(λ) is the number of parts of λ, and the crank of partition λ, the crank c(λ) of a partition is defined as