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%I #12 Apr 07 2020 22:36:34
%S 0,1,0,4,12,56,2,1360,2672,320,16,350000,431232,107904,12864,3872,320,
%T 255036992,234667968,98251776,19523136,10633728,1615552,1182720,
%U 163520,127360,13440
%N Triangle T(d,v) read by rows: the number of hyper-tetrahedra with volume v/d! defined by selecting d+1 vertices of the d-dimensional unit-hypercube.
%C The unit hypercube in dimension d has 2^d vertices, conveniently expressed by their Cartesian coordinates as binary vectors of length d of 0's and 1's. Hyper-tetrahedra (simplices) are defined by selecting a subset of 1+d of them. The (signed) volume V of a tetrahedron is the determinant of the d vectors of the edges divided by d!. (The volume may be zero if some edges in the tetrahedron are linearly dependent.) The triangle T(d,v) is a histogram of all A136465(d+1) tetrahedra classified by absolute (unsigned) volume V=v/d!.
%C The number of non-flat simplices (row sums without the leftmost column) are tabulated by Brandts et al. (Table 1, column beta_n). - _R. J. Mathar_, Feb 06 2016
%H J. Brandts, Sander Dijkhuis et al, <a href="http://arxiv.org/abs/1209.3875">There are only two nonobtuse binary triangulations of the unit n-cube</a>, arXiv:1209.3875 and <a href="http://dx.doi.org/10.1016/j.comgeo.2012.09.005">Comput. Geometry 46 (2013) 286</a>
%e In d=2, 4 tetrahedra (triangles) are defined by taking subsets of d+1=3 vertices out of the 2^2=4 vertices of the unit square. Each of them has the same volume (area) 1/2!, so T(d=2,v=1)=4.
%e In d=3, 12 = T(d=3,v=0) tetrahedra with zero volume are defined by taking subsets of d+1=4 vertices out of the 2^3=8 vertices of the unit cube. These are the cases of taking any 4 vertices on a common face. (There are 6 faces and two different edge sets for each of them; one with edges along the cube's edges, and one with edges along the face diagonals.)
%e The triangle starts in row d=1 as follows:
%e 0 1;
%e 0 4;
%e 12 56 2;
%e 1360 2672 320 16 ;
%e 350000 431232 107904 12864 3872 320;
%Y Cf. A136465 (row sums), A003432 (maximum column index), A004145 (column v=0).
%K nonn,tabf
%O 1,4
%A _R. J. Mathar_, Feb 04 2016
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