Jörg Rambau has authored 3 sequences.
A363512
The number of affine dependencies among the vertices of the n-cube.
Original entry on oeis.org
1, 20, 1348, 353616, 446148992, 2118502178496, 38636185528212416
Offset: 2
For n = 2, there is only one affine dependence among the vertices of the square involving all points.
For n = 3, since there are 6 embeddings of the square into the boundary and 6 embeddings of the square into the interior of the 3-cube, there are 12 affine dependences on squares; moreover, there is an affine dependence for each of the 8 vertices of the 3-cube coming from the intersection of the line from that vertex to the vertex opposite in the 3-cube with the triangle spanned by the neighbors of that vertex; this adds up to a total of 20 affine dependencies.
Cf.
A363506 for the same numbers up to symmetry. Related to
A007847 (and
A363505, resp.) by oriented-matroid duality.
A363506
The number of affine dependencies among the vertices of the n-cube up to symmetry.
Original entry on oeis.org
1, 3, 15, 186, 12628, 3591868, 3858105362
Offset: 2
For n = 2, all vertices of the square constitute the only affine dependence.
For n = 3, there is an affine dependence in each boundary square all of which are equivalent; moreover, there is one affine dependence in each square cutting the cube in half all of which are equivalent; the remaining affine dependence with five elements contains a triangle spanned by all neighbors of a point together with that point and the point opposite to it in the 3-cube.
Cf.
A363512 for the total numbers (not up to symmetry). Related to
A363505 (and
A007847, resp.) by oriented-matroid duality.
A363505
Number of hyperplanes spanned by the vertices of an n-cube up to symmetry.
Original entry on oeis.org
2, 3, 6, 15, 63, 623, 22432, 3899720
Offset: 2
For n = 2, it can be seen that there are only two non-equivalent hyperplanes spanned by vertices of the square: one spanned by a boundary edge having all remaining points on one side and one spanned by a diagonal separating the remaining points.
For n = 3, we again have a hyperplane parallel to a coordinate plane spanned by a boundary square having all the remaining points on one side; moreover, a hyperplane spanned by the four points on the opposite axis-parallel parallel boundary edges of two opposite boundary squares leaving two remaining points on either side, and a skew hyperplane spanned by the three neighbors of a single point separating that point from the remaining points.
A007847 gives the total numbers (not up to symmetry). Related to
A363506 (and
A363512, resp.) by oriented-matroid duality.
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