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Counting permutations of $X^2$ that induce 4 quasigroup operations up to isotopy

Let $X$ be a finite set. Recall that a binary operation $\ast$ on $X$ is said to be a quasigroup operation if there are binary operations $/,\backslash$ where $(x/y)\ast y=(x\ast y)/y=x$ and $x\ast(x\...
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