In how many different ways can we place N identical rooks on a chess board WITHOUT CORNERS so that no two of them attack each other (for N > 3)? I'm trying, unsuccessfully, to figure it out for a moment now
For N = 8, the chessboard would look like: 

Just for reference here is a similar question without the corner restriction and with N = 8 : In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?

This problem can be conveniently solved using rook polynomials. We can pairwise exchange rows and columns without changing the number of possible arrangements of non-attacking rooks. An equivalent board is shown below.
                                                                
The rook polynomial of the four forbidden squares in the top left corner is
\begin{align*}
1+4x+2x^2\tag{1}
\end{align*}
where the coefficient of $x^k$ gives the number of ways to place $k$ non-attacking rooks on the forbidden squares. The number of valid arrangements to place eight non-attacking rooks on the $(8\times 8)$ board respecting the forbidden squares is given by (1) using the inclusion-exclusion principle as
\begin{align*}
1\cdot 8!-4\cdot 7!+2\cdot 6!\color{blue}{=21\,600}
\end{align*}
We can place eight non-attacking rooks in $8!$ ways, subtract $4\cdot7!$ ways having at least one rook on one of the four forbidden squares and add as compensation for double counting  $2\cdot6!$ ways having two non-attacking rooks on the four forbidden squares.
