Let $G$ be a digraph (not necessarily a DAG) and let $s,t \in V(G)$. What is the complexity of counting the number of simple $s-t$ paths in $G$.
I would expect the problem to be #${\mathsf P}$-complete but have not been able to locate an exact reference.
Also notice that a number of similar questions have been answered correctly here and elsewhere but not this precise question - to emphasise I am not interested in counting walks and/or undirected graphs (in the first case the variant is in ${\mathsf P}$ and in the other #${\mathsf P}$-hard).