Partitions weighted by the parity of the crank

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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 Me(n) minus the number of partitions of n with odd crank Mo(n). We also discuss the combinatorial implications of q-series identities involving Me(n)Mo(n). Finally, we determine the exact values of Me(n)Mo(n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

Keywords

Partitions
Crank
Congruences

Cited by (0)

The first author was supported by the Korea Research Foundation Grant funded by the Korean government (KRF-2008-331-C00005), the second author was supported by the SRC program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) (R11-2007-035-01002-0), and the third author was supported by an ACI “Jeunes Chercheurs et Jeunes Chercheuses”.