Number of transitive relations on a set Let $T(n)$ denote the number of transitive binary relations on an $n$-element set. So T(1) = 2 and T(2) = 13, for of the 16 possible relations on a 2-element set {a,b}, the only three which are not transitive are
(i) {(a,b), (b,a)},
(ii) {(a,a), (a,b), (b,a)},
(iii) {(b,b), (a,b), (b,a)}.
There is some literature on this function - a wikipedia entry, a sequence in Sloane up to $n=18$, a few research papers on this and similar kinds of functions, including some complicated formulas for $T(n)$. However, I have not seen anywhere a "nice asymptotic estimate" for $T(n)$. Using the data in Sloane I computed, for $1 \leq n \leq 18$, the function $f(n) = \frac{\log_{2} T(n)}{n^2}$, obtaining the following approximate values 
1, 0.9251, 0.8242, 0.7477, 0.6894, 0.6435, 0.6063, 0.5755, 0.5494, 0.5270, 0.5075, 0.4903, 0.4751, 0.4614, 0.4491, 0.4380, 0.4278, 0.4184
So my general question is whether anything (non-trivial) is known about the asymptotics of $T(n)$, and a more specific question is whether $f(n) \rightarrow 0$ as $n \rightarrow  \infty$ ?     
 A: If $P(n)$ is the number of partial orders, then $\log_2 P(n) = n^2/4 + o(n^2)$, an old result of Kleitman. Look in MathSciNet for many different sharpenings. Now if $T(n)$ is the number of  transitive relations, then Klaska proved that $T(n)$ and $2^n P(n)$ are asymptotically equal.  Therefore,  $\log_2 T(n) = n^2/4 + o(n^2)$.  Ref to Klaska: MR1446401 (98c:05006) 
Klaška, Jiří. Transitivity and partial order. Math. Bohem. 122 (1997), no. 1, 75–82.
A: Yes, a lot is known. Transitive relations are the same (essentially; there is a slight problem with  self-loops) as strongly connected digraphs. See:
http://www.math.uwaterloo.ca/~nwormald/papers/dicores.pdf
and references therein.
EDIT oops, answers a wrong (but related) question... Formulas (though not asymptotics) for the quantities in question appear in
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.6879&rep=rep1&type=pdf
Another Edit Partial orders (as mentioned by @Aaron) have been asymptotically enumerated, and the log is asymptotic to $n^2/4.$ (see Kleitman, D. J.; Rothschild, B. L.
Asymptotic enumeration of partial orders on a finite set. 
Trans. Amer. Math. Soc. 205 (1975), 205–220. )
