Intermediate Problems between FP and #P

Question

Do there exist intermediate problems (in the sense of Ladner's Theorem) for FP vs. #P? I assume that something is known, because I read some papers concerned with FP/#P dichotomies. However, I couldn't find a reference.

Answer 1

Use Schöning's theorem:

Let $A_1$, $A_2$ be recursive sets and $C_1$, $C_2$ be classes of recursive sets with the following properties:

1. $A_1 \notin C_1$, $A_2 \notin C_2$
2. $C_1$ and $C_2$ are recursively presentable,
3. $C_1$ and $C_2$ are closed under finite variations.

Then there exists a recursive set $A$ such that:

1. $A \notin C_1$, $A \notin C_2$,
2. if $A_1 \in \mathsf{P}$ and $A_2\notin \{ \emptyset, \Sigma^* \}$, then $A \leq^{\mathsf{P}}_m A_2$.

For the purposes of counting dichotomy theorems, the two relevant classes of decision problems are $\text{P}$ and $\text{P}^{\#\text{P}}$.