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A288177 Maximum number of vertices of any convex polygon formed by drawing all line segments connecting any two lattice points of an n X m convex lattice polygon in the plane written as triangle T(n,m), n >= 1, 1 <= m <= n.

%I A288177 #35 Aug 06 2024 11:05:24
%S A288177 3,4,4,4,4,4,4,4,5,5,4,5,5,6,6,4,5,5,6,6,6,4,5,6,6,6,7,7,4,5,7,6,7,7,
%T A288177 7,7,4,5,6,6,7,7,8,8,8,4,5,6,6,7,7,8,8,8,7,4,5,6,6,7,7,8,8,8,8,8,4,5,
%U A288177 7,6,7,7,8,7,8,8,8,8,4,5,8,6,7,7,8,7,8,8,8,8,8,4,5,8,6,7,7,8,8,8,8,8,8,8,8,4,5,8,6,7,7,8,8,8,8,8,8,9,9,9,4,5,7,6,7,7,8,8,8,8,8,8,9,9,9,9,4,5,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,4,5,8,7,8,8,8,8,8,8,8,8,9,9,9,10,10,9
%N A288177 Maximum number of vertices of any convex polygon formed by drawing all line segments connecting any two lattice points of an n X m convex lattice polygon in the plane written as triangle T(n,m), n >= 1, 1 <= m <= n.
%C A288177 The table is given in the section "Results" of the notes by M. E. Pfetsch and G. M. Ziegler, see link.
%C A288177 An n X m convex lattice polygon presumably means an n X m grid of square cells, formed using a grid of n+1 X m+1 points. - _N. J. A. Sloane_, Feb 07 2019
%H A288177 Huntington Tracy Hall, <a href="https://citeseerx.ist.psu.edu/pdf/e5d8674f2073ad215df390ea390d802103ae6cea">Counterexamples in Discrete Geometry</a>. Dissertation UC, Berkeley, Fall 2004.
%H A288177 Serkan Hosten, Diane Maclagan, and Bernd Sturmfels, <a href="https://arxiv.org/abs/math/0105036">Supernormal Vector Configurations</a>, arXiv:math/0105036 [math.CO], 4 May 2001.
%H A288177 Marc E. Pfetsch and Günter M. Ziegler, <a href="http://www.mathematik.tu-darmstadt.de/~pfetsch/chambers/">Large Chambers in a Lattice Polygon</a> (Notes), March 28, 2001, December 13, 2004.
%H A288177 Marc E. Pfetsch and Günter M. Ziegler, <a href="/A288177/a288177_1.pdf">Large Chambers in a Lattice Polygon</a> (Notes), March 28, 2001, December 13, 2004. [Cached copy, with permission]
%H A288177 Hugo Pfoertner, <a href="/A288177/a288177.pdf">Illustrations of Chamber Complexes up to 5 X 5</a>.
%e A288177 Drawing the diagonals in a lattice square of size 1 X 1 produces 4 triangles, so T(1,1)=3.
%e A288177 Triangle begins:
%e A288177   3;
%e A288177   4, 4;
%e A288177   4, 4, 4;
%e A288177   4, 4, 5, 5;
%e A288177   4, 5, 5, 6, 6;
%e A288177   4, 5, 5, 6, 6, 6;
%e A288177   4, 5, 6, 6, 6, 7, 7;
%e A288177   ...
%Y A288177 Cf. A288178 (diagonal of table), A288179, A288180, A288181, A288187.
%K A288177 nonn,tabl
%O A288177 1,1
%A A288177 _Hugo Pfoertner_, Jun 06 2017