Number of relations on a set

Question

What is the number of relations on a $n$ element set that are antisymmetric and not symmetric? I have soved this question using the fact that 'antisymmetric and not symmetric' means asymmetric... So answer will be $$3^{\left(\frac{n^2-n}{2}\right)}$$ Is it correct?

Answer 1

No -- $3^{(n^2-n)/2}$ counts the relations that are antisymmetric and irreflexive (it also counts the relations that are antisymmetric and reflexive, but that is unlikely to be what you had in mind). Instead:

First let's count the antisymmetric relations. For each of the $\binom n2$ unordered pairs or elements we can either have $aRb$ or $bRa$ or neither, giving 3 choices. Additionally each of the $n$ elements can either be related to itself or not.

One of these relations is symmetric exactly if all of the $\binom n2$ answers are "neither" (and then it is immaterial which elements are related to themselves). So you need to subtract those cases.