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

Question

If a set has $n$ elements, how many transitive relations are there on it?

For example if set $A$ has $2$ elements then how many transitive relations. I know the total number of relations is $16$ but how to find only the transitive relations? Is there a formula for finding this or is it a counting problem?

Also how to find this for any number of elements $n$?

Answer 1

There is no simple formula for this number (but see http://oeis.org/A006905 for the values for small $n$). The case $n=2$ is small enough that you can list out all 16 different relations and count the ones that are transitive. (You should get 13 of them.)

Answer 2

As noticed by @universalset, there are 13 transitive relations among a total of 16 relations on a set with cardinal 2. And here are they :)

Answer 3

Counting transitive relations on a set is probably very hard. Just recently, in this paper of mine titled, "On the number of transitive relations on a set", I was able to find several recursive relations and lower bounds for the number of transitive relations on a set.