The problem involves arranging $n$ rooks on an $n \times n$ chessboard such that their placement is symmetric about the main diagonal (the diagonal from the lower left to the upper right) and no two rooks attack each other. Let's analyze the possible formulas:

Key observations:

1. Symmetry about the diagonal: If a rook is placed at position $(i, j)$, another rook must be placed at $(j, i)$ to maintain symmetry.
2. Non-attacking condition: No two rooks can be in the same row or column.

Understanding the structure:

• The positions of the rooks can be described as permutations of the set $\{1, 2, \dots, n\}$ such that each number appears exactly once in the sequence, representing rows and columns.
• To ensure diagonal symmetry, the placement must follow the constraint that $(i, j)$ is paired with $(j, i)$. This means a rook on $(i, j)$ forces a placement at $(j, i)$.

Recursive analysis:

To compute $R_n$, consider:

1. Base cases:

   • $R_1 = 1$: One rook fits on a $1 \times 1$ board.
   • $R_2 = 1$: There is only one way to place two rooks symmetrically on a $2 \times 2$ board.

2. Recursion relation:

   • When placing a rook at position $(i, i)$, there are $n$ choices for $i$.
   • The remaining positions reduce the problem to placing $n-1$ rooks on an $(n-1) \times (n-1)$ board.
   • Additionally, off-diagonal placements may pair with $(i, j)$ and $(j, i)$, introducing terms involving $R_{n-2}$.

The correct recursive relation is: $R_n = n \cdot R_{n-1} + (n-1) \cdot R_{n-2}.$

This corresponds to the fourth formula: $R_n = n \cdot R_{n-1} + (n-1) \cdot R_{n-2}.$

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Final Answer:

The correct formula is: $R_n = n \cdot R_{n-1} + (n-1) \cdot R_{n-2}.$

Would you like a step-by-step derivation or examples to illustrate this?

Related Questions:

1. What are the explicit values of $R_3$ and $R_4$ based on the recurrence relation?
2. How does the diagonal symmetry condition restrict the arrangement of rooks?
3. Can $R_n$ be expressed in terms of factorials or combinatorial coefficients?
4. What is the growth rate of $R_n$ as $n \to \infty$?
5. How does the problem change if we consider symmetry about other diagonals?

Tip:

To solve problems with recursive formulas, always analyze base cases and carefully account for all contributing terms in the recurrence relation.