Partitions weighted by the parity of the crank

Abstract

The ‘crank’ is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank M e ( n ) minus the number of partitions of n with odd crank M o ( n ) . We also discuss the combinatorial implications of q-series identities involving M e ( n ) − M o ( n ) . Finally, we determine the exact values of M e ( n ) − M o ( n ) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.