In how many different ways can k bishops be placed on an nxn chessboard such that no two bishops attack each other? Please try to respond with a formula and explanation.

Call this number $B_k(n)$. For fixed $k$ it is known that $B_k(n)$ has the form $P_k(n)+(1)^nQ_k(n)$, where $P_k$ and $Q_k$ are polynomials. These polynomials have been computed for (at least) $k\leq 6$. We also have (unsurprisingly) the asymptotic formula $B_k(n)\sim n^{2k}/k!$. For further information see http://www.math.binghamton.edu/zaslav/Tpapers/bishops.slides.20100729.pdf. Update. I learned from Tom Zaslavsky that an explicit formula for $B_k(n)$ as a triple sum was given by C. E. Arshon in 1936. 


See my book http://problem64.beda.cz/silo/kotesovec_non_attacking_chess_pieces_2013_6ed.pdf , page 234236. Number of ways to place k nonattacking bishops on an n x n board, (n>1) 

