The complexity of counting simple paths in a directed graph
The #P-completeness proof of counting simple s-t paths in both undirected and directed graphs can be found in:

Leslie G. Valiant: The Complexity of Enumeration and Reliability Problems. SIAM J. Comput. 8(3): 410-421 (1979)

From the paper:

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4. Some #P-complete problems
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14. S-T PATHS (i.e. SELF AVOIDING WALKS) (directed or undirected)
Input: $G; s,t \in V$
Output: Number of (directed) paths from $s$ to $t$ that visit every node at most once.
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