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Monotonic Matrix


A monotonic matrix of order n is an n×n matrix in which every element is either 0 or contains a number from the set {1,...,n} 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 n=1, 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].

The maximum numbers of cells occupied in an n×n matrix for n=1, 2, ... are given by 1, 2, 5, 8, 11, 14, 19, ... (OEIS A070214).


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References

Sloane, N. J. A. Sequences A070214 and A086976 in "The On-Line Encyclopedia of Integer Sequences."Stein, S. K. and Szabó, S. Algebra and Tiling. Washington, DC: Math. Assoc. Amer., p. 95, 1994.

Referenced on Wolfram|Alpha

Monotonic Matrix

Cite this as:

Weisstein, Eric W. "Monotonic Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MonotonicMatrix.html

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