= Solution
For each unordered pair $\{a,b\}$ with $a\ne b$, <antisymmetry> allows three choices: neither direction, only $(a,b)$, or only $(b,a)$. This contributes $3^{\binom n2}$. The diagonal is unrestricted by <antisymmetry>, but being not <irreflexive> requires a nonempty <subset> of the $n$ diagonal pairs. There are $2^n-1$ such choices. Hence
$$
\boxed{(2^n-1)3^{\binom n2}.}
$$
This allows some diagonal pairs to be absent; “not <irreflexive>” does not require a <reflexive relation>.
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