Counting the number of distinct s-t cuts in a oriented graph
Counting the number of s,t-cuts is #P-complete.  This is a result by Provan and Ball.


J. Scott Provan, Michael O. Ball: The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput. 12(4): 777-788 (1983)


Therefore, unless some complexity-theoretic collapse happens, you cannot get essentially faster algorithm than listing all of them.