This chess position ranking wouldn't be possible without making use of such combinatorics, as you can see with the use [1] [2] of the multinomial ranking [3].

This chess position ranking wouldn't be possible without making use of such combinatorics, as you can see with the use [1] [2] of the multinomial ranking [3]. You're right that counting is much easier than ranking. That's reflected in my Haskell counting program [4] being MUCH simpler and MUCH faster than the ranking program. But counting is only good for upper bounding and doesn't let you draw a random sample to determine the fraction of legal ones, as needed for estimation. [1] https://github.com/tromp/ChessPositionRanking/blob/main/src/Chess/Position/Ranking.hs#L412 https://github.com/tromp/ChessPositionRanking/blob/main/src/... [2] https://github.com/tromp/ChessPositionRanking/blob/main/src/Chess/Position/Ranking.hs#L424 https://github.com/tromp/ChessPositionRanking/blob/main/src/... [3] https://github.com/tromp/ChessPositionRanking/blob/main/src/Data/Ranking.hs#L14-L37 https://github.com/tromp/ChessPositionRanking/blob/main/src/... [4] https://github.com/tromp/ChessPositionRanking/blob/main/src/CountChess.hs https://github.com/tromp/ChessPositionRanking/blob/main/src/...