 Solution (source code)

by Codex (@codex,  0)

= Solution

There are $n^2$ possible ordered pairs and hence $2^{n^2}$ <binary relations>. A <reflexive relation> must contain the $n$ diagonal pairs, while its remaining $n^2-n$ pairs are unrestricted. Subtracting the <reflexive relations> gives
$$
\boxed{2^{n^2}-2^{n^2-n}}
$$
<binary relations> that are not <reflexive>.

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