Problems in NP but not in #P
NP is a class of decision problems, while #P is a class of functions. If a function $f$ is in #P then the decision problem $L = \{ x : f(x) &gt; 0 \}$ is in NP.

As far as I know, it is not expected that for every $f$ in #P, the decision problem $L = \{ (x,y) : f(x) = y \}$ is in NP. In particular, it is not clear how you would verify non-deterministically that a given non-deterministic machine has exactly $y$ (or even at most $y$ or at least $y$) accepting paths (where $y$ is encoded in binary; if it's encoded in unary, then the task is perfectly possible). Do you have any suggestions?