A070214 Maximal number of occupied cells in all monotonic matrices of order n.

1, 2, 5, 8, 11, 14, 19, 23, 28, 32, 38, 43, 49, 55

Offset: 1

Author

N. J. A. Sloane, Jul 24 2003, Jun 19 2007

Keywords

• nonn
• more
• hard
• nice

Comments

A monotonic matrix of order n is an n X n matrix in which every cell contains 0 or 1 numbers from the set {1...n} subject to 3 conditions:

(1) The filled-in entries in each row are strictly increasing;

(2) The filled-in entries in each column are strictly decreasing;

(3) For two filled-in cells with same entry, the one further right is higher (the positive slope condition).

From Rob Pratt: The problem can be formulated as a maximum independent set problem in a graph with n^3 nodes (i, j, k) in {1, 2, ..., n}^3. If node (i, j, k) appears in the solution, the interpretation is that cell (i, j) should contain k. The arcs, which indicate conflicting choices, are as follows:

Arc joining (i1, j1, k1) and (i2, j2, k2) if:

[rows increasing] i1 = i2 and ((j1 < j2 and k1 >= k2) or (j1 > j2 and k1 <= k2)).

[columns decreasing] j1 = j2 and ((i1 < i2 and k1 <= k2) or (i1 > i2 and k1 >= k2)).

[one color per cell] i1 = i2 and j1 = j2 and k1 <> k2.

[positive slope] k1 = k2 and i1 <> i2 and (j2 - j1) / (i2 - i1) > 0.

Examples

			a(3) >= 5 from this matrix:
  2 - 3
  - - 1
  1 3 -
a(5) >= 11 from this matrix:
  - - 4 - 5
  4 - - 5 -
  - - 1 2 3
  3 5 - - -
  1 2 - - -
Dean Hickerson found the following matrix, which improves the lower bound for a(8) to 23: (This is now known to be optimal)
  - - 2 - - 4 7 8
  - - 1 7 8 - - -
  7 8 - - - - - -
  - 2 - 4 - - - 6
  - 1 - - - 3 6 -
  4 - - - 6 - - -
  2 - - - 3 - - 5
  1 - - 3 - - 5 -
Paul Jungeblut improves the lower bound for a(11) to 38 with this matrix.
  -- --  8 -- -- --  9 -- -- -- 11
  --  8 -- -- --  9 -- -- -- -- 10
   8 -- -- --  9 -- -- -- 10 11 --
  -- -- -- -- -- -- -- --  4  5  7
  --  4 -- -- --  5  7 11 -- -- --
  -- -- -- -- --  1 -- --  2  3  6
   4 -- -- --  5 --  6 10 -- -- --
  -- -- -- --  1 --  2  3 -- -- --
   2  3  7 11 -- -- -- -- -- -- --
  --  1  6 10 -- -- -- -- -- -- --
   1 --  5  9 -- -- -- -- -- -- --
		

Links

• 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.
• W. Hamaker and S. K. Stein, Combinatorial packing of R^3 by certain error spheres, IEEE Trans. Information Theory, 30 (No. 2, 1984), 364-368.
• Patric R. J. Östergård, and Antti Pöllänen, New Results on Tripod Packings, Discrete & Computational Geometry 61.2 (2019): 271-284
• S. K. Stein and S. Szabo, Algebra and Tiling, MAA Carus Monograph 25, 1994, page 95.
• Alexandre Tiskin, Tripods do not pack densely, Lecture Notes in Computer Science, 1858 (2000), 272-280.
• Alexandre Tiskin, Packing tripods: narrowing the density gap, Discrete Math., 307 (2007), 1973-1981.
• Eric Weisstein's World of Mathematics, Monotonic Matrix

Crossrefs

Cf. A086976.

Formula

a(r*s) >= a(r)*a(s); if a(n) = n^e(n) then L := lim_{n->infinity} e(n) exists and is in the range 1.513 <= L <= 2.

Tiskin showed that a(n) = o(n^2).

Extensions

a(1)-a(5) computed by K. Joy. a(6) = 14 was established by Szabo.

Jul 27 2003 - Aug 23 2003: Rob Pratt has used integer programming to confirm the values for n <= 6 and has shown that a(7) = 19, 23 <= a(8) <= 28, 28 <= a(9) <= 42 and 32 <= a(10) <= 62.

Extended to a(14) from Tiskin (2007), who gives a(15) >= 61, a(16) >= 65.

a(11) corrected by Paul Jungeblut, Jul 09 2020