The number of ways to place k bishops in a nonattacking configuration on an n by n square board is a well known result and can for example be found in http://problem64.beda.cz/silo/kotesovec_non_attacking_chess_pieces_2013_6ed.pdf

However, trying to generalize the result to a rectangular board seems rather difficult. Has there been any progress on this front?

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    $\begingroup$ Can't the black squares and the white squares be thought of separately, that is, sum over $w+b=k$ bishops, and place $b$ black, non-attacking bishops on the board... Then, the black squares and black bishops can be thought of as rooks, and then you are dealing with rook placements, for which there is extensive literature. $\endgroup$ – Per Alexandersson Aug 29 '15 at 18:10
  • $\begingroup$ Yes, that is the normal method of dealing with the problem for the square case, in which the squares of each color form a Ferrers board. However, in the general rectangular case, you only have a quasi-Ferrers board, which makes the calculation more difficult, and as far as I know, there is no literature on rook polynomials for such boards. $\endgroup$ – ruadath Aug 30 '15 at 11:23

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