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 $n \rightarrow \infty$ ? 



Number of transitive relations on a set
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.


