Monotonic matrix

In combinatorics, a monotonic matrix of size n is a square matrix of size n with entries in the set of integers ${\displaystyle \{0,1,\dots ,n\}}$ such that

1. the nonzero entries in each row are strictly increasing from left to right,
2. the nonzero entries in each column are strictly decreasing from top to bottom, and
3. (positive slope condition) for two nonzero cells of the same entry, the one further right is placed higher than the early one.

Equivalently, a square matrix with integer entries is monotonic if the corresponding semicrosses are disjoint.[1]

An example is:

${\displaystyle {\begin{bmatrix}2&&3\\&&1\\1&3&\\\end{bmatrix}}}$.

References

1. Stein & Szabó, Ch. 4, § 1., Exercise 11. harvnb error: no target: CITEREFSteinSzabó (help)

• Stein, S. K. and Szabó, S. Algebra and Tiling. Washington, DC: Math. Assoc. Amer., p. 94, 1994.
• Boris Aronov, Vida Dujmović, Pat Morin, Aurélien Ooms, Luís Fernando Schultz Xavier da Silveira, More Turán-Type Theorems for Triangles in Convex Point Sets, arXiv:1706.10193 [math.CO], 2017.

Further reading

• Sloane, N. J. A. Sequences A070214 and A086976 in "The On-Line Encyclopedia of Integer Sequences."