Weak Queens

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 $S(14)$.

Searching for $S(14)$ or the total number of distinct configurations for placing weak queens on a $14 \times 14$ chessboard is a complex and computationally intensive problem. In fact, this question does not lend itself to a straightforward analytical approach, and thus requires a combinatorial search algorithm to generate all valid placements of queens and then count them. This kind of problem is more suitable for solution with computer programming than it is for manual mathematics.

Here is a skeleton of a principle how one might solve this problem with programming:

1. **Initialize the board**: Create an $n \times n$ matrix to simulate the chess board.

2. **Define the rules**: Implement the behaviour of the weak queen as defined in the problem statement.

3. **Determine valid placements**: Develop a recursive function to place queens on the board in a systematic way that abides by the rules of the weak queen’s movements.

4. **Count the configurations**: Each time a valid configuration (i.e. a configuration where no two queens are threatening each other) is identified, increment a counter.

5. **Sum up the answers**: After generating all configurations for a given weakness factor, sum all the valid configurations obtained for each weakness factor up to $n-1$.

Writing and running this program would require appropriate programming skills and computational resources. Additionally, depending on the specific algorithm’s efficiency and the computational resources available, finding $S(14)$ may take a significant amount of time due to the explosive growth of possible configurations as $n$ increases.

In conclusion, the precise value of $S(14)$ cannot be easily determined without resorting to computational methods as it involves thorough combinatoric search which is non-trivial for particularly larger boards like $14 \times 14$.

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