%I #13 Jun 08 2023 10:57:38

%S 1,3,15,186,12628,3591868,3858105362

%N The number of affine dependencies among the vertices of the n-cube up to symmetry.

%C a(n) is also the number of circuits of any point configuration combinatorially equivalent to a unit cube in dimension n up to symmetry.

%H Jörg Rambau, <a href="https://www.wm.uni-bayreuth.de/de/team/rambau_joerg/TOPCOM/SymLexSubsetRS.pdf">Symmetric lexicographic subset reverse search for the enumeration of circuits, cocircuits, and triangulations up to symmetry</a>, Manuscript distributed with <a href="https://www.wm.uni-bayreuth.de/de/team/rambau_joerg/TOPCOM/">TOPCOM</a>.

%e For n = 2, all vertices of the square constitute the only affine dependence.

%e 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.

%Y Cf. A363512 for the total numbers (not up to symmetry). Related to A363505 (and A007847, resp.) by oriented-matroid duality.

%K nonn,hard,more

%O 2,2

%A _Jörg Rambau_, Jun 06 2023