A monotonic matrix of order 
is an 
matrix in which every element is either 0 or contains
a number from the set 
subject to the conditions
1. The filled-in elements in each row are strictly increasing,
2. The filled-in elements in each column are strictly decreasing, and
3. Positive slope condition: for two filled-in cells with same element, the one further right is in an earlier row.
The numbers of distinct monotonic matrices of orders 
, 2, ... are 2, 19, 712, ... (OEIS A086976).
For example, the monotonic matrices of order 2 are
![[0 0; 0 0],[0 0; 0 1],[0 0; 0 2],[0 0; 1 0],[0 0; 1 2],[0 0; 2 0],
[0 1; 0 0],[0 1; 1 0],[0 1; 2 0],[0 2; 0 0],[0 2; 0 1],[0 2; 1 0],
[0 2; 2 0],[1 0; 0 0],[1 0; 0 2],[1 2; 0 0],[2 0; 0 0],[2 0; 0 1],
[2 0; 1 0].](/images/equations/MonotonicMatrix/NumberedEquation1.svg) |
The maximum numbers of cells occupied in an 
matrix for 
, 2, ... are given by 1, 2, 5, 8, 11, 14, 19, ... (OEIS A070214).
