# Amount of transitive relations on a finite set

• How many transitive relations on a set of $n$ elements?
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.