Nonattacking configurations of $k$ bishops on an $m$ by $n$ rectangular board The number of ways to place $k$ bishops in a nonattacking configuration on an $n$ by $n$ square board is a known and can for example be found in http://problem64.beda.cz/silo/kotesovec_non_attacking_chess_pieces_2013_6ed.pdf (Czech/English).
However, trying to generalize the result to a rectangular board seems rather difficult. Has there been any progress on this front?
 A: Search
Berghammer R. Relational Modelling and Solution of Chessboard Problems. in: RAMICS
(Relational and algebraic methods in computer science), 12th international conference,
Rotterdam, Netherlands May 20 - June 3 2011. Heidelberg, Springer; 2011
If I'm not mistaken, Dr Berghammer investigated the topic at this conference, although I didn't go, so double check it.
I believe that the max # of ind bishops is either $m+n-1$ or $m+n-2$ depending upon three cases, with $n>m$:  $m,n$ both odd; exactly one of $m,n$ are even, and $m,n$ both even.  So, the results are $m+n-2$ for $m=n$ or $m,n$ both even and $n>m$; otherwise $m+n-1$.  It has been a while since I worked through the proofs of this independently, but as for the formations:
Given an $m \times n$ board, with $n>m$, plant bishops in the first and last columns.  This will leave squares in the center that are neither attacked nor occupied, if $n>m+1$.  You can then plant the needed number of independent bishops in the center.
Apologies for not being formal.  My independent proof (which I've since deleted) was discovered sometime ago, and I don't feel it necessary to go through in detail considering the info is out there and the proof is a bit long.
I might add that the number of ways of placing maximum independent sets of bishops on the rectangular board is open.  Kotosevec did find the number of ways of placing $k$ independent bishops on the square board.
