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Let me elaborate on the use of Pólya enumeration for this problem. Let $G\simeq S_m \times S_n$ act on $X=\{x_{i,j}\colon 1\leq i \leq m, 1\leq j \leq n\}$ via $(\sigma,\pi)\cdot x_{i,j} = x_{\sigma(i),\pi(j)}$, and via this action view $G\subseteq S_{mn}$. For $g\in G$ let $c_i(g)$ be the number of $i$-cycles of $g$ (in this embedding into $S_{mn}$), and ...

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