 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 $3^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}$. 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. HenceThis allows some diagonal pairs to be absent; “not irreflexive” does not require a reflexive relation.