A simple question on #P
Expanding on Timot's comment, $\# P$ consists of all functions which can be expressed as the number of accepting computations of some non-deterministic Turing machine running in polynomial time. It is also the class of all functions which can be expressed at the number of non-accepting computations of some non-deterministic Turing machine running in polynomial time (exercise).

If $f \in \# P$ then $\{ x : f(x) \geq 1 \} \in N\! P$ and $\{x : f(x) = 0 \} \in \mathrm{co}N\! P$; and vice versa, if $L \in N\!P$ ($L \in \mathrm{co}N\!P$) then it can be expressed in the form $\{ x : f(x) \geq 1 \}$ ($\{ x : f(x) = 0 \}$) for some $f \in \# P$.