The Idea is this:
The Decision problems Max Cut for graphs is APX-Hard and therefore does not have a PTAS.
However Dense versions of Max Cut do have a PTAS!
This suggest that Counting Perfect Matchings for Dense Bipartite Graphs could have some sort of speed up.
Does such a speedup exists in the complexity literature?
Further to the above, I am aware of the expectional work done on NonNegative Matrices. However I was really thinking about General matrices and expecting exponential algorithms. Of course #P is by standard complexity theory intractable!