In how many ways can I arrange \(32\) black and \(32\) white pawns on an \(8 \times 8\) chessboard such that no pawn attack another?

- A black pawn can attack only a white pawn, and vice versa.
- The usual chess rules of attacking follow, i.e, white pawns attack squares immediately diagonally in front, and the black pawns attack squares immediately diagonally backward (as seen by the white observer).
- All rotation, reflections count as distinct.

For reference:

- There are \(3\) such configurations on a \(2 \times 2\) board with \(2\) white and black pawns each.
- There are \(30\) such configurations on a \(4 \times 4\) board with \(8\) white and black pawns each.

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