Given $N$, I would like to know the number of matrices constructed from $1$ to $N$ which satisfy the following conditions:
1. Each row entry should be in increasing order.
2. Each column entry should be in increasing order.
For example: when $N = 4$, there are 4 matrices which satisfy these conditions:
1. $\left( \begin{array}{c} 1 & 2 & 3 & 4\end{array} \right) $
2. $\left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4\end{array} \right) $
3. $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4\end{array} \right) $
4. $\left( \begin{array}{cc} 1 & 3 \\ 2 & 4\end{array} \right) $
My observations:
$N$ should be a composite number to construct a valid matrix.
If a matrix $A$ satisfies the condition then $A^T$ also satisfies the condition.