Intermediate Problems between FP and #P
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:
  
  
  $A_1 \notin C_1$, $A_2 \notin C_2$
  $C_1$ and $C_2$ are recursively presentable, 
  $C_1$ and $C_2$ are closed under finite variations. 
  
  
  Then there exists a recursive set $A$ such that:
  
  
  $A \notin C_1$, $A \notin C_2$, 
  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 $\text{P}^{\#\text{P}}$.