Similar question: How can one determine the chess configuration that maximizes the number of possible moves?
Say you have an $8 \times 8$ chessboard with only 8 white rooks (don't want to worry about capturing).
What is the best possible configuration (best in the sense that it maximizes the total number of possible moves on move 1)? I believe it is is with rooks all along the diagonals. This way, each rook can move $7$ units total horizontally and $7$ units total vertically as no piece is obstructing any other. So it will be $(2 \times 7) \times 8$.
Now, another obvious possible configuration is to just put the rooks on the other diagonal. Are there other configurations? Is there a way to set this problem up with group theory in a more general setting where we have some fixed number & type of pieces, and wish to count the number of initial configurations maximizing the possible movesets?