How many transitive and symetric relations that are not equivalence are in a set of $n$ elements?

Question

I have a Set $S$, $|S|=n$, and I need to count how many symetric and transitive relations are in $S$ that are not equivalence relations.

I know how to count equivalence relations (Bell number) but I don't konw how to count the relations that are symetric and transitive at the same time.

Answer 1

There are $B_n$ equivalence relations on $[n]$. As shown in the two threads linked to in the comments, there are $B_{n+1}$ relations on $[n]$ that are symmetric and transitive. Thus there are $B_{n+1}-B_n$ relations that are symmetric and transitive but not reflexive.