Partitions weighted by the parity of the crank

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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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Number Theory, Combinatorics, 11P81, 11P82, 11P83, 05A17, 33D15, 11F11

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http://salesiana.dossiersoluciones.com/handle/123456789/125422

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0703266
 http://arxiv.org/abs/math/0703266

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arXiv