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A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.

%I A333543 #31 Nov 06 2020 08:47:55
%S A333543 1,4,72,24,162816,96576,118464,64896,45888,22464,19776,11904,8640,
%T A333543 8448,6144,1728,1152,384,384,384
%N A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.
%C A333543 Rows 1 through 4 computed by _Veit Elser_, later confirmed by _Tom Karzes_.
%C A333543 The row sums give A333539.
%D A333543 N. J. A. Sloane, Cutting Up a Cube, Math Fun Mailing List, Apr 10 2020; with replies from _Tom Karzes_, _Tomas Rokicki_, _Veit Elser_, and others.
%H A333543 Veit Elser, <a href="/A333539/a333539.txt">Rows 1 through 4</a>
%H A333543 Scott R. Shannon, <a href="/A331452/a331452_6.png">Illustration for a(2) = 4.</a>
%H A333543 Scott R. Shannon, <a href="/A333543/a333543.png">Illustration for a(3) = 72</a>. This shows the 4-faced cells in the 3D cube dissection. The 72 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin.
%H A333543 Scott R. Shannon, <a href="/A333543/a333543_1.png">Illustration for a(4) = 24</a>. This shows the 5-faced cells in the 3D cube dissection. The 24 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin. These polyhedra form a perfect octahedron inside the original cube with its points touching the cube's inner surface.
%e A333543 The two diagonals of a square cut it into four triangular pieces, so T(2,3) = 4.
%e A333543 Triangle begins:
%e A333543 1,
%e A333543 4,
%e A333543 72, 24,
%e A333543 162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384,
%e A333543 ...
%Y A333543 Cf. A333539, A333540, A333544, A338622 (number of k-faced polyhedra for the 3D Platonic solids).
%Y A333543 For the number of hyperplanes see A007847.
%K A333543 nonn,tabf,more
%O A333543 1,2
%A A333543 _N. J. A. Sloane_, Apr 21 2020