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.



How to place k bishops on an nxn chessboard
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. 


