TSTST 2024 Problem 9

Let $n \geq 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from 1 to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$ .