OFFSET
1,2
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.
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.
Sherman K. Stein and Sándor Szabó, 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.
Alexander Tiskin, Packing tripods: narrowing the density gap, Discrete Math., 307 (2007), 1973-1981.
Eric Weisstein's World of Mathematics, Monotonic Matrix
Wikipedia, Tripod packing.
FORMULA
a(r*s) >= a(r)*a(s); if a(n) = n^e(n) then L := lim_{n->oo} e(n) exists and is in the range 1.513 <= L <= 2.
Tiskin showed that a(n) = o(n^2).
EXAMPLE
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. (Also optimal, per Östergård and Pöllänen.)
-- -- 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 -- -- -- -- -- -- --
CROSSREFS
KEYWORD
nonn,more,hard,nice,changed
AUTHOR
N. J. A. Sloane, Jul 24 2003, Jun 19 2007
EXTENSIONS
a(1)-a(5) computed by K. Joy. a(6) = 14 was established by Szabó.
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
Removed a(12)-a(14) from Tiskin (2007) because his optimal values are for 3-symmetric packings only. - Andrei Zabolotskii, Jul 14 2026
STATUS
approved