Valiant defined $\#P$ in terms of a counting TM, which is a NTM that outputs the number of solutions .
I am a bit stuck with the following two questions: Let's say I have a decision problem $X$, the corresponding counting problem $\#X$, and an enumeration algorithm $E$ that enumerates the solutions of $X$ in polynomial output complexity.
- If $w$ is a witness for $X$ that I can verify in polynomial time, does this imply that $\#X$ is in $\#P$?
- Does the existence of $E$ imply that $\#X$ is in $\#P$?
For both questions, I think the answer is yes because they imply that there is a NTM which could be modified to count the number of witnesses. However, I feel like I cannot argue this properly and that I might miss something.
 Leslie G. Valiant: The Complexity of Computing the Permanent. Theor. Comput. Sci. 8: 189-201 (1979)