What is in #P and what is not?

Understand

For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not.

• We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations.
• We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints.
• This is directly linked to classical combinatorial proofs of algebraic identities and inequalities.