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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. 1
0, 1, 0, 4, 12, 56, 2, 1360, 2672, 320, 16, 350000, 431232, 107904, 12864, 3872, 320, 255036992, 234667968, 98251776, 19523136, 10633728, 1615552, 1182720, 163520, 127360, 13440 (list; graph; refs; listen; history; text; internal format)
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!.
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
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.
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.)
The triangle starts in row d=1 as follows:
0 1;
0 4;
12 56 2;
1360 2672 320 16 ;
350000 431232 107904 12864 3872 320;
Cf. A136465 (row sums), A003432 (maximum column index), A004145 (column v=0).
Sequence in context: A149419 A367450 A151469 * A092576 A149420 A298680
R. J. Mathar, Feb 04 2016

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Last modified February 29 07:23 EST 2024. Contains 370414 sequences. (Running on oeis4.)