Is there a way to estimate the number of possible checkmates on a chess board? By possible, I mean that it corresponds to some board state. If it's easier to estimate, I don't mind whether the number of pieces is appropriate or whether the board state is actually reachable, although I think that would be more interesting if possible.
If possible, an approximate density for the number of checkmate states out of the state space considered would be interesting also.
I've seen some upper bounds for chess board states (e.g., here and here), so one possible idea I had might be to first consider a subset of board configurations based on restrictions like the possible number of pieces, and then create a random sample of board states within the considered space. You could then create a confidence interval for the possible density, and from there calculate a confidence interval for the possible number of checkmate states. But I'm not sure how to appropriately create such a sample, or whether it's even possible.