Eccoci.

The classical eight queens puzzle is the well known problem of placing eight chess queens on an $8 \times 8$ chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of $92$ distinct configurations can be found for eight queens. The general case asks for the number of distinct ways of placing $n$ queens on an $n \times n$ board, e.g. you can find $2$ distinct configurations for $n=4$.

Let's define a weak queen on an $n \times n$ board to be a piece which can move any number of squares if moved horizontally, but a maximum of $n - 1 - w$ squares if moved vertically or diagonally, $0 \le w \lt n$ being the "weakness factor". For example, a weak queen on an $n \times n$ board with a weakness factor of $w=1$ located in the bottom row will not be able to threaten any square in the top row as the weak queen would need to move $n - 1$ squares vertically or diagonally to get there, but may only move $n - 2$ squares in these directions. In contrast, the weak queen is not handicapped horizontally, thus threatening every square in its own row, independently from its current position in that row.

Let $Q(n,w)$ be the number of ways $n$ weak queens with weakness factor $w$ can be placed on an $n \times n$ board so that no two queens threaten each other. It can be shown, for example, that $Q(4,0)=2$, $Q(4,2)=16$ and $Q(4,3)=256$.

Let $S(n)=\displaystyle\sum_{w=0}^{n-1} Q(n,w)$.

You are given that $S(4)=276$ and $S(5)=3347$.

Find $S(14)$.