Some properties of partitions in terms of crank

Eki̇n, A.

doi:10.1090/S0002-9947-00-02306-0

Some properties of partitions in terms of crank
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by A. Bülent Eki̇n

Trans. Amer. Math. Soc. 352 (2000), 2145-2156

DOI: https://doi.org/10.1090/S0002-9947-00-02306-0

Published electronically: February 16, 2000

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Abstract:

Let $N(r,m,n)$ (resp. $M(r,m,n))$ denote the number of partitions of $n$ whose ranks (resp. cranks) are congruent to $r$ modulo $m$. Atkin and Swinnerton-Dyer gave the relations between the numbers $N(r,m,mn+k)$ when $m=5,~7$ and $0\leq r,k<m$. Garvan gave the relations between the numbers $M(r,m,mn+k)$ when $m=5,7$, and $11$, $0\leq r,k <m$. Here, we show that the methods of Atkin and Swinnerton-Dyer can be extended to prove the relations for the crank.

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