We are given a partially ordered set $P$. Let $L$ denote the set of all linear extensions of $P$ (or equivalently, the set of all topological sortings of the nodes).
We want to count the number of labeled DAGs which give rise to any of those linear extensions $L$.
For example, we are given a set of nodes $\{A,B,C\}$ and that $A < B$. The set of linear extensions $L$ is $\{ABC, ACB, CAB \}$. The number of DAGs with 3 nodes is 25, out of which 9 have a path from $A$ to $B$.
Has this problem been studied before? I was not able to find anything about it.
A related problem is that of counting the number of linear extensions of a given partially ordered set. This problem is #P-complete. Intuitively, my problem seems to be as hard as the one of counting linear extensions.
Edit: I probably gave a wrong formulation of my problem. I reformulated it to better match what I had in mind. I hope I got it right now.