On the rank and the crank modulo 4

Lewis, Richard

doi:10.1090/S0002-9939-1991-1057957-2

On the rank and the crank modulo $4$
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Proc. Amer. Math. Soc. 112 (1991), 925-933 Request permission

Abstract:

Let $N(r,m,n)$ (respectively, $M(r,m,n)$) denote the number of partitions of $n$ whose ranks (respectively, cranks) are congruent to $r$ modulo $n$. It is shown that $N(0,4,2n + 1) = M(1,4,2n + 1)$ and $N(2,4,2n) = M(1,4,2n)$.

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Some properties of

the number of partitions of