Summary (take from Math Reviews):
Dyson defined the rank of a partition as the largest part minus the number of parts. Let $N(m,t,n)$ denote the number of partitions of $n$ of rank congruent to $m\bmod t$. Dyson conjectured and Atkin and Swinnerton-Dyer proved that $N(m,5,5n+4)=\frac 15p(5n+4)$, $0 \leq m\leq 4$; $N(m,7,7n+5)=\frac 17p(7n+5)$, $0\leq m\leq 6$, where $p(n)$ denotes the number of partitions of $n$. Dyson also observed that the rank did not separate the partitions of $11n+6$ into $11$ equal classes although Ranumujan's conjecture $p(11 n+6)\equiv 0\bmod 11$ holds. He conjectured the existence of crank for the proof of this conjecture. Garvan defined the crank relative to certain vector partitions of $n$ to prove all these conjectures. In this article the authors define the crank of an ordinary partition of $n$ denoted by $c(\pi)$. Let $N_V (m,n)$ denote the number of vector partitions of $n$ with crank $m$. It is proved that the number of partitions of $n$ with $c(\pi)=m$ is equal to $N_V(m,n)$ for all $n>1$.