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COMMENTS
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These terms count all legal chess piece arrangements on the given rank (row). In addition to the restrictions noted in the Sleek link, pawns are never counted as being on ranks 1 and 8 (Pawns begin a game on the player's second rank, can never move or capture backwards, and upon reaching the player's eighth rank are promoted immediately to either a Knight, Bishop, Rook, or Queen). Disregarded for these counts are whether White or Black has the next move, facts related to move numbers or repetitions of position, availability of castling or en passant captures, and game outcome.
There are 214358881, 302752688, 65081202 arrangements with no Kings, one King, two Kings, respectively, for each of ranks 2 through 7. The corresponding numbers are 43046721, 73089744, 18220674 for each of rank 1 and 8.
Because Forsyth-Edwards Notation (FEN) for a single rank (ignoring the parts after all 8 ranks are described) has a one-to-one correspondence with chess rank diagrams, this sequence answers Sleek's question.
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