Partitions weighted by the parity of the crank

Abstract. A partition statistic ‘crank’ gives combinatorial interpretations for Ramanujan’s famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of partitions with even crank Me(n) minus the number of partitions with odd crank Mo(n) satisfies. For example, we show that Me(5n + 4)− Mo(5n + 4) ≡ 0 (mod 5). We also determine the exact values of Me(n) − Mo(n) in case of partitions into distinct parts, which are at most two and zero for infinitely many n.