Example 4.17 Find the rook polynomial $R(t)$ for an $n \times n$ chessboard.
Solution: From Example 4.16, $r_{k}=\binom{n}{k} \cdot(n)_{k}$, so
$$
R(t)=\sum_{k=0}^{\infty} r_{k} t^{k}=\sum_{k=0}^{n}\binom{n}{k}(n)_{k} t^{k} .
$$

For convenience, use the symbol “ $\times$ ” to replace the square “ $\square$ ” on the chessboard.