Suppose we have a random $DAG(n, p)$. Here is how it's generated:
Put n distinct nodes on a line, and connect each node in the $i$th order to any node after that; This would form a complete directed graph with $n$ nodes.
Sample each edge with probability $p$.
The resulting graph will be directed, and acyclic.
The question is: what is the number of paths of length $\ell$ in a random $DAG(n, p)$?
Note: I know the answer for an undirected graph.