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For generic (not necessarily symmetric) $m\times n$ matrices over a set of $k$ elements, the number of those with pairwise distinct columns and rows is $$\sum_{i=0}^m\sum_{j=0}^n s(m,i)\cdot s(n,j)\cdot k^{i\cdot j},$$ where $s(,)$ are Stirling numbers of first kind with sign.