cp's OEIS Frontend

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

%I A070214 #33 Feb 16 2025 08:32:46
%S A070214 1,2,5,8,11,14,19,23,28,32,38,43,49,55
%N A070214 Maximal number of occupied cells in all monotonic matrices of order n.
%C A070214 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:
%C A070214 (1)  The filled-in entries in each row are strictly increasing;
%C A070214 (2)  The filled-in entries in each column are strictly decreasing;
%C A070214 (3)  For two filled-in cells with same entry, the one further right is higher (the positive slope condition).
%C A070214 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:
%C A070214 Arc joining (i1, j1, k1) and (i2, j2, k2) if:
%C A070214 [rows increasing] i1 = i2 and ((j1 < j2 and k1 >= k2) or (j1 > j2 and k1 <= k2)).
%C A070214 [columns decreasing] j1 = j2 and ((i1 < i2 and k1 <= k2) or (i1 > i2 and k1 >= k2)).
%C A070214 [one color per cell] i1 = i2 and j1 = j2 and k1 <> k2.
%C A070214 [positive slope] k1 = k2 and i1 <> i2 and (j2 - j1) / (i2 - i1) > 0.
%H A070214 Boris Aronov, Vida Dujmović, Pat Morin, Aurélien Ooms, Luís Fernando Schultz Xavier da Silveira, <a href="https://arxiv.org/abs/1706.10193">More Turán-Type Theorems for Triangles in Convex Point Sets</a>, arXiv:1706.10193 [math.CO], 2017.
%H A070214 W. Hamaker and S. K. Stein, <a href="https://doi.org/10.1109/TIT.1984.1056868">Combinatorial packing of R^3 by certain error spheres</a>, IEEE Trans. Information Theory, 30 (No. 2, 1984), 364-368.
%H A070214 Patric R. J. Östergård, and Antti Pöllänen, <a href="https://doi.org/10.1007/s00454-018-0012-2">New Results on Tripod Packings</a>, Discrete & Computational Geometry 61.2 (2019): 271-284
%H A070214 S. K. Stein and S. Szabo, <a href="https://www.maa.org/press/ebooks/algebra-and-tiling">Algebra and Tiling</a>, MAA Carus Monograph 25, 1994, page 95.
%H A070214 Alexandre Tiskin, <a href="https://doi.org/10.1007/3-540-44968-X_27">Tripods do not pack densely</a>, Lecture Notes in Computer Science, 1858 (2000), 272-280.
%H A070214 Alexandre Tiskin, <a href="https://doi.org/10.1016/j.disc.2004.12.028">Packing tripods: narrowing the density gap</a>, Discrete Math., 307 (2007), 1973-1981.
%H A070214 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MonotonicMatrix.html">Monotonic Matrix</a>
%F A070214 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.
%F A070214 Tiskin showed that a(n) = o(n^2).
%e A070214 a(3) >= 5 from this matrix:
%e A070214   2 - 3
%e A070214   - - 1
%e A070214   1 3 -
%e A070214 a(5) >= 11 from this matrix:
%e A070214   - - 4 - 5
%e A070214   4 - - 5 -
%e A070214   - - 1 2 3
%e A070214   3 5 - - -
%e A070214   1 2 - - -
%e A070214 Dean Hickerson found the following matrix, which improves the lower bound for a(8) to 23: (This is now known to be optimal)
%e A070214   - - 2 - - 4 7 8
%e A070214   - - 1 7 8 - - -
%e A070214   7 8 - - - - - -
%e A070214   - 2 - 4 - - - 6
%e A070214   - 1 - - - 3 6 -
%e A070214   4 - - - 6 - - -
%e A070214   2 - - - 3 - - 5
%e A070214   1 - - 3 - - 5 -
%e A070214 Paul Jungeblut improves the lower bound for a(11) to 38 with this matrix.
%e A070214   -- --  8 -- -- --  9 -- -- -- 11
%e A070214   --  8 -- -- --  9 -- -- -- -- 10
%e A070214    8 -- -- --  9 -- -- -- 10 11 --
%e A070214   -- -- -- -- -- -- -- --  4  5  7
%e A070214   --  4 -- -- --  5  7 11 -- -- --
%e A070214   -- -- -- -- --  1 -- --  2  3  6
%e A070214    4 -- -- --  5 --  6 10 -- -- --
%e A070214   -- -- -- --  1 --  2  3 -- -- --
%e A070214    2  3  7 11 -- -- -- -- -- -- --
%e A070214   --  1  6 10 -- -- -- -- -- -- --
%e A070214    1 --  5  9 -- -- -- -- -- -- --
%Y A070214 Cf. A086976.
%K A070214 nonn,more,hard,nice
%O A070214 1,2
%A A070214 _N. J. A. Sloane_, Jul 24 2003, Jun 19 2007
%E A070214 a(1)-a(5) computed by K. Joy. a(6) = 14 was established by Szabo.
%E A070214 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.
%E A070214 Extended to a(14) from Tiskin (2007), who gives a(15) >= 61, a(16) >= 65.
%E A070214 a(11) corrected by _Paul Jungeblut_, Jul 09 2020