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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.