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Partitions weighted by the parity of the crank

Dohoon Choi·Soon-Yi Kang·Jeremy Lovejoy
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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 $M_e(n)$ minus the number of partitions with odd crank $M_o(n)$ satisfies. For example, we show that $M_e(5n+4)-M_o(5n+4)\equiv 0 \pmod 5.$ We also determine the exact values of $M_e(n)-M_o(n)$ in case of partitions into distinct parts, which are at most two and zero for infinitely many $n$.

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Dohoon Choi, Soon-Yi Kang, Jeremy Lovejoy. 2007-03-09. Partitions weighted by the parity of the crank. https://arxiv.org/abs/math/0703266

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