ELMO SL 2013 C4

Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be Muirhead-able if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$ , the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$ . For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing?

Proposed by Evan Chen