Consider an $n \times n$ chessboard. It is required to move a rook from the lower left corner to the upper right corner. You can only move up and to the right, without stepping on the cells of the main diagonal and below it. (The rook is on the main diagonal only at the initial and final moments.) How many such routes does the rook have?
Establish the correspondence between all such rook routes and the sequences of numbers from problem 60447.

## Solution

Let the rook's first move be upwards, then the last move must be to the right. In addition, it must make $n-1$ moves upwards and the same number to the right. Correspond each move upwards to a one, and each move to the right to a minus one. We will obtain a sequence of $2 n-2$ numbers that satisfies the conditions of problem 60447. Conversely, from each such sequence, we can construct a rook's route in the upper part of the board.

## Answer

$2 C_{n-1}$.