<s>[INST] How do I calculate how many ways 14 non-attacking bishops can be placed on a chessboard? It's possible to place up to 14 non-attacking bishops on a chessboard.
How can I calculate the number of valid configurations, without writing a program to brute force it?
I've checked Non Attacking Chess Pieces, but it doesn't cover the variation I'm interested in, and a quick Google hasn't yielded useful results.
UPDATE: 
Proof that only 14 bishops can be placed:
If we divide the chessboard into diagonals, and we treat diagonal 1 and 15 as the same diagonal, then the maximum number of bishops per diagonal is 1, therefore 14.

Here is an example configuration of 14 bishops to show it's possible:

NOTE: Not a duplicate of this question because I am asking about the number of arrangements of 14 bishops, not the maximum number of bishops on a chess board.
 [/INST] Let's convert the chessboard into points: (rank,file). The normal chessboard refers to file A through H. We will use 1 through 8. Consider the number rank minus file:
$$\begin{matrix} & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 8 & | &  7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ 7 & | & 6 & 5 & 4 & 3 & 2 & 1 & 0 & -1 \\ 6 & | & 5 & 4 & 3 & 2 & 1 & 0 & -1 & -2 \\ 5 & | & 4 & 3 & 2 & 1 & 0 & -1 & -2 & -3 \\ 4 & | & 3 & 2 & 1 & 0 & -1 & -2 & -3 & -4 \\ 3 & | & 2 & 1 & 0 & -1 & -2 & -3 & -4 & -5 \\ 2 & | & 1 & 0 & -1 & -2 & -3 & -4 & -5 & -6 \\ 1 & | & 0 & -1 & -2 & -3 & -4 & -5 & -6 & -7\end{matrix}$$
Notice how these numbers are constant on the diagonals?
Next consider rank + file:
$$\begin{matrix} & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 8 & | & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 7 & | & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 6 & | & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ 5 & | & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ 4 & | & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 3 & | & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 2 & | & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 1 & | & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\end{matrix}$$
Notice how these numbers are constant on the opposite diagonals? So, you are looking for 14 pairs of numbers from 1 through 8 such that you get 14 distinct pairs (rank-file,rank+file) such that no other pair shares the same rank-file or rank+file.
I went through my old notes to see where I remember a similar problem to this. It may have been the seminar I took that discussed block designs. I did not actually take a class that focused on block designs, so I do not recall the set up. I think this approach could be interesting (and give more details about configurations), but also far more time consuming and maybe not worth it. Specifically, this is a problem for a Transversal Design.
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