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

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 $n$?

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.)
• It's easier to just find the three nontransitive relations on $\{a,b\};$; they are $\{(a,b),(b,a)\}\cup S$ where $S$ is a proper subset of $\{(a,a),(b,b)\}.$ – bof Feb 28 '17 at 8:48
• @bof: That's true for size $n = 2$ but not true for any sizes past that. E.g., for $n = 3$, only 33% are transitive; for $n = 4$, 6%; for $n = 5$, 0.5%; etc. – Daniel R. Collins Nov 11 '18 at 3:03
Although there's no formula, results for small $n$ can be obtained by recursion. This paper proves that, if there are $T_n$ transitive relations and $P_n$ partial orders on an $n$-element set, and if we define $N_k\left( n\right):=\sum_{s=0}^k\binom{n}{s}S\left( n-s,\,k-s\right)$ where $S\left( n,\,k\right)=\frac{1}{k!}\sum_{i=1}^k\left( -1\right)^{k-i}\binom{n}{k}i^n$, then $T_n = \sum_{k=1}^n N_k\left( n\right)P_k$. Unfortunately, $P_n$ is also only known for small $n$; we can't obtain further $P_n$ with any known recursion, which in turn caps computing $T_n$.