We know that #SAT is #P-complete. We also know that problems with polynomial decision versions like PERMANENT are #P-complete. Is it true that finding the number of simple cycles in a graph, i.e. #CYCLE is also #P-complete? This is also a problem with decision version solved in polynomial time.
I suppose that it is either not-known or false, because it might be used a a popular example in textbooks like Arora and Barak's book, etc.