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?



How many idempotent relations are there on an $n$-element set?
The answer seems to be in Butler, K. K.-H.,The number of idempotents in (0,1)-matrix semigroups, Linear Algebra and Its Applications 5 (1972), 233–246. I will see if I have access to the journal and will tell you more.


**Edit.** The paper is [here](http://ac.els-cdn.com/0024379572900055/1-s2.0-0024379572900055-main.pdf?_tid=8eece726-80e4-11e2-9fcb-00000aab0f26&acdnat=1361973171_ff59deefabd771cffb05e77de54d3a1e) for free. It counts the idempotents by D-class so it is not written down in a simple succinct formula. If you google idempotent boolean matrix there are further papers which may be of use.


