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.
Based on the specified graph construction scheme, a path of length $k$ requires $k+1$ vertices, in ascending vertex number order.
It follows that there are exactly ${\large{\binom{n}{k+1}}}$ potential paths of length $k$.
For each such potential path, the probability that it's an actual path is $p^k$.
Therefore, the expected number of paths of length $k$ is ${\large{\binom{n}{k+1}}}p^k$.