cp's OEIS Frontend

A268408 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.

%I A268408 #12 Apr 07 2020 22:36:34
%S A268408 0,1,0,4,12,56,2,1360,2672,320,16,350000,431232,107904,12864,3872,320,
%T A268408 255036992,234667968,98251776,19523136,10633728,1615552,1182720,
%U A268408 163520,127360,13440
%N A268408 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 A268408 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 A268408 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 A268408 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 A268408 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 A268408 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 A268408 The triangle starts in row d=1 as follows:
%e A268408 0 1;
%e A268408 0  4;
%e A268408 12  56  2;
%e A268408 1360  2672  320  16 ;
%e A268408 350000  431232  107904  12864  3872  320;
%Y A268408 Cf. A136465 (row sums), A003432 (maximum column index), A004145 (column v=0).
%K A268408 nonn,tabf
%O A268408 1,4
%A A268408 _R. J. Mathar_, Feb 04 2016