Question 40
Let $n \geq 1$ be a positive integer. A square table of dimensions $n \times n$ is filled arbitrarily with the numbers $1, 2, \ldots, n^2$, so that every number appears exactly once in the table. From each row, select the smallest number, and let the greatest of them be denoted by $x$. From each column, select the greatest number, and let the smallest of them be denoted by $y$. The table is called equilibrated if $x = y$. How many equilibrated tables exist?