The number of unique undirected graphs on $n$ labeled vertices without multiple edges or self-loops is $2^{n \choose 2}$ as the accepted answer to this question suggests. I would like to know how many of those graphs satisfy transitive closure.
I've been trying to enumerate them all for small number of vertices to get some intuition and I believe for graphs on 3 labeled vertices, there 8 possible undirected graphs (no self-loops and no multiple edges), 5 of which satisfy transitive closure. For 4 vertices, there are 64 possible graphs, 15 of which satisfy transitive closure.
How can I get a closed-form expression that counts the number of unique undirected graphs on labeled vertices with no self-loops and no multiple-edges that satisfy transitive closure?
