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The number of antisymmetric relations is still relatively easy to find. I don’t know why you say the diagonal has to be in the relation; the usual definition doesn’t require this. Under the usual definition, the diagonal is unconstrained, yielding a factor $2^n$, and each off-diagonal pair has three possibilities, yielding a factor $3^{n(n-1)/2}$, for a total of $2^n3^{n(n-1)/2}$ antisymmetric relations. This is OEIS sequence A083667.
The number of transitive relations is indeed more difficult to determine. This is OEIS sequence A006905. The OEIS page cites this article, which obtains a recurrence relation by establishing a bijection with partial orders.
Quite generally, for this sort of sequence the best way to find what’s known about it is to calculate the first few terms and use them to locate the sequence in OEIS.