%I #19 Feb 17 2024 10:26:08

%S 1,4,23,180,1806,20198

%N Triangle partitions of order n: topologically distinct ways to dissect a triangle into n triangles.

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/triangle.html">Triangles</a>

%H Z. Skupien, A. Zak, Pair-sums packing and rainbow cliques, in <a href="http://www.math.uiuc.edu/~kostochk/Zykov90-Topics_in_Graph_Theory.pdf">Topics In Graph Theory</a>, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144, (in English and Russian).

%H Miroslav Vicher, <a href="http://www.vicher.cz/puzzle/triangles/triangles.htm">Triangle Partitions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangleDissection.html">Triangle Dissection</a>

%e From _M. F. Hasler_, Feb 15 2024: (Start)

%e a(2) = 1 because up to equivalence, there is only one partition of a triangle in two smaller ones, using a segment from one vertex to a point on the opposite side. (Here and below, "on" excludes the endpoints.)

%e a(3) = 4 is the number of partitions of a triangle ABC into three smaller ones: One uses three segments AD, BD and CD, where D is a point inside ABC. Three other topologically inequivalent partitions of order 3 each use two segments, as follows: {AE, AF}, {AE, EG} and {AE, BH}, where E and F are two distinct points on BC, G is a point on AB, and H is a point on AE. (End)

%Y Cf. A053740.

%K nonn,more,nice,hard

%O 2,2

%A _N. J. A. Sloane_, Sep 01 2000