The rank of a partition was a concept introduced by Freeman Dyson in 1944 to provide a combinatorial explanation for the Ramanujan congruences \(5k+4\equiv 0(\mod 5)\) and \(7k+5\equiv 0(\mod 7)\). The rank of a partition $$ n=\lambda_k+\lambda_{k-1}+\dots + \lambda_2+\lambda_1,\qquad \lambda_k\geq \lambda_{k-1}\geq \dots \geq \lambda_2\geq \lambda_1 $$ is defined to be \(\lambda_k - k\), namely, the largest part minus the number of parts. If \(N(m,q,n)\) denotes the number of partitions of \(n\) with rank \( \equiv m (\mod q)\), what Dyson conjectured was that for each integer \(k\geq 0\), $$ N(0,5,5k+4)=N(1,5,5k+4)=N(2,5,5k+4)=N(3,5,5k+4)=N(4,5,5k+4). $$ Thus, all the partitions of \(5k+4\) could be separated into five equinumerous subsets, and hence \(5|5k+4\). Dyson similarly conjectured that \begin{multline*} N(0,7,7k+5)=N(1,7,7k+5)=N(2,7,7k+5)=N(3,7,7k+5)\\ =N(4,7,7k+5)=N(5,7,7k+5)=N(6,7,7k+5). \end{multline*} Both conjectures were subsequently proved by A. O. L. Atkin and H. P. F. Swinnerton-Dyer in a 1954 paper.
To illustrate the truth of Dyson's first conjecture, the following table groups the partitions of \(9=5(1)+4\) by rank, modulo 5: $$ \begin{array}{c|c|c|c|c} \text{partition}&\text{largest part}, \lambda_k& \text{number of parts}, k& \text{rank }=\lambda_k -k&\text{rank }(\mod 5)\\ \hline \{2,2,1,1,1,1,1\} & 2 & 7 & -5 & 0 \\ \{3,3,3\} & 3 & 3 & 0 & 0 \\ \{4,2,2,1\} & 4 & 4 & 0 & 0 \\ \{4,3,1,1\} & 4 & 4 & 0 & 0 \\ \{5,1,1,1,1\} & 5 & 5 & 0 & 0 \\ \{7,2\} & 7 & 2 & 5 & 0 \\ \{2,2,2,1,1,1\} & 2 & 6 & -4 & 1 \\ \{3,1,1,1,1,1,1\} & 3 & 7 & -4 & 1 \\ \{4,3,2\} & 4 & 3 & 1 & 1 \\ \{4,4,1\} & 4 & 3 & 1 & 1 \\ \{5,2,1,1\} & 5 & 4 & 1 & 1 \\ \{8,1\} & 8 & 2 & 6 & 1 \\ \{1,1,1,1,1,1,1,1,1\} & 1 & 9 & -8 & 2 \\ \{2,2,2,2,1\} & 2 & 5 & -3 & 2 \\ \{3,2,1,1,1,1\} & 3 & 6 & -3 & 2 \\ \{5,2,2\} & 5 & 3 & 2 & 2 \\ \{5,3,1\} & 5 & 3 & 2 & 2 \\ \{6,1,1,1\} & 6 & 4 & 2 & 2 \\ \{3,2,2,1,1\} & 3 & 5 & -2 & 3 \\ \{3,3,1,1,1\} & 3 & 5 & -2 & 3 \\ \{4,1,1,1,1,1\} & 4 & 6 & -2 & 3 \\ \{5,4\} & 5 & 2 & 3 & 3 \\ \{6,2,1\} & 6 & 3 & 3 & 3 \\ \{9\} & 9 & 1 & 8 & 3 \\ \{2,1,1,1,1,1,1,1\} & 2 & 8 & -6 & 4 \\ \{3,2,2,2\} & 3 & 4 & -1 & 4 \\ \{3,3,2,1\} & 3 & 4 & -1 & 4 \\ \{4,2,1,1,1\} & 4 & 5 & -1 & 4 \\ \{6,3\} & 6 & 2 & 4 & 4 \\ \{7,1,1\} & 7 & 3 & 4 & 4 \\ \end{array} $$ From the table it can be seen that $$ N(0,5,9)=N(1,5,9)=N(2,5,9)=N(3,5,9)=N(4,5,9) = 6. $$

Let the number of partitions of \(n\) with rank \(m\) be denoted by \(N(m, n)\). The generating function for the number of partitions of \(n\) with rank \(m\) is \[ \sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}N(m,n)z^mq^n =\sum_{k=0}^{\infty}\frac{q^{k^2}}{(zq,q/z;q)_k}. \] If \(Q(m,n)\) denotes the number of partitions of \(n\) into distinct parts with rank \(m\), then the generation function is \[ \sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}Q(m,n)z^mq^n =\sum_{k=0}^{\infty}\frac{q^{k(k+1)/2}}{(zq;q)_k}. \]

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