Number of different positions of rooks on chessboard I know that this topic as been mentioned before, but no accurate answer has been provided.
Suppose we have to place $n$ rooks on $n \times n$ chessboard so that no one attacks another. How to count the number of different ways to place them up to rotations of the chessboard?
I have been trying the Burnside's lemma but did not obtain satisfactory results...
 A: It seems that the first complete solution to this kind of problem was worked out by Édouard Lucas in section 128 of his textbook on number theory:


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*Édouard Lucas, Théorie des Nombres (1891)


Notice that he considers solutions different up to rotations and reflections, so you have to slightly modify the argument. Here is an outline:
You begin with the $n!$ total solutions, and consider the group of symmetries $\Gamma$ of the chessboard you are interested in. Those are:
$\Gamma_1$: Identity
$\Gamma_2$:Rotation by $\pi$ radians.
$\Gamma_3$:Rotation by $\pm \pi/2$ radians.
If we denote $\sigma_n$ the number of differente solutions (under those symmetries), then $\sigma_n$ is the number of orbits of $\Gamma$ on the set of the $n!$ solutions, so that you can now apply Burnside's lemma.
Now, $\Gamma_1=n!$ and $\Gamma_2=(n/2)!2^{n/2}$.
Also $\Gamma_3=(2m)!/m!$ if $n=4m$ or $n=4m+1$, and $\Gamma_3=0$ otherwise.
From here you can easily count the number of solutions.
For a clearer and more modern exposition of the solution considering also reflections, you can consult:


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*Robert W. Robinson, Counting arrangements of bishops (1976)

