Combinatorics: how many ways can we mark nodes in a DAG as black or white, if every node downstream of a black node is also black?
Your problem is #P-complete. The set of solutions can be identified with the set of antichains with the poset corresponding to the DAG (the correspondence identifies a coloring with the black vertices that have no black parents), counting the number of which has been shown to be #P-complete by Provan and Ball in The complexity of counting cuts and of computing the probability that a graph is connected.