As far as I know, it is an open problem to give a formula counting transitive relations on an $n$-element set. Is it easier to count the idempotent relations, that is relations that are both transitive and interpolative? (A relation $\rho$ is interpolative when $x\rho y\implies((\exists z)\ x\rho z \wedge z\rho y).$)

Also, if we denote the number of transitive relations on an $n$-element set by $T_n$ and the number of idempotent relations by $I_n$, can we say what the asymptotic behavior of $I_n/T_n$ is?