Let #EXP be the counting variant of NEXP, in the same way that #P is the counting variant of NP. Are there any known #EXP-complete problems? In particular, has #Succinct Sat (the natural candidate) been shown to be #EXP-complete?
Papadimitriou and Yannakis mention this class in a 1986 paper (1) but I have not been able to find more recent results.
As an aside, they mention an interesting natural candidate for a #EXP problem: given a number $n$ in binary, return the number of planar graphs with $n$ nodes. Is anything further known about the complexity of this problem?
- Papadimitriou, Christos H., and Mihalis Yannakakis. "A note on succinct representations of graphs." Information and Control 71.3 (1986): 181-185.