SIMON-MARAIS 2018 B4

The following problem is open in the sense that no solution is currently known. An explicit expression with proof for the known case of $|A(5, n)|$ will be awarded 2 points. Further progress on the problem may be awarded more points.

For positive integers $m$ and $n$, let $A(m, n)$ be the set of $2 \times mn$ matrices $M$ with entries from the set $\{1, 2, \ldots, m\}$ such that

• each of the numbers $1, 2, \ldots, m$ appears exactly $2n$ times;
• $M_{1,1} \leq M_{1,2} \leq \cdots \leq M_{1,mn}$ and $M_{2,1} \leq M_{2,2} \leq \cdots \leq M_{2,mn}$; and
• $M_{1,j} < M_{2,j}$ for $j = 1, 2, \ldots, mn$.

Determine $|A(m, n)|$.