I am wondering what is known about the problem of finding the number of solution to a NP-complete problem. We can of course take SAT as an example, it doesn't matter that much. It is clear that this is harder than finding a solution to the problem and (equivalently) knowing if there is a solution or not. The problem that I see is that the number of solution may be very big, therefore any strategy of finding one, removing it with a clause and iterating is doomed.
Still it feels like this problem might be in the realm of NP problems. Indeed, the SAT that seems to be hard to handle are those that have a lot of solutions or very few solutions, which are not in general the hardest SATs. I was thinking about something like checking if for some variable $x_i$, one of $x_i\rightarrow C$, $\lnot x_i\rightarrow C$, $C\rightarrow x_i$ and $C\rightarrow\lnot x_i$ is true, which leads to a big reduction/augmentation in the number of solutions, then we can iterate using another pivot variable, but I do not know what is the worst case for this, probably something that is such that none of those are true every time we ask a question, which means that the number of solution is essentially moderate.
Anyways I would be interested in any literature on the subject, I couldn't find any.