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Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.
6

%I #6 Mar 09 2024 11:15:02

%S 0,1,66,920,6350,29505,106036,317856,832140,1961025,4248310,8590296,

%T 16398746,29814785,51983400,87399040,142333656,225359361,347978730,

%U 525376600,777308070,1129138241,1613050076,2269437600

%N Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

%C Each member of a chiral pair is a reflection, but not a rotation, of the other.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

%F a(n) = (n-1) * n^2 * (n+1) * (8 - 5*n^2 + n^4) / 48.

%F a(n) = 1*C(n,2) + 63*C(n,3) + 662*C(n,4) + 2400*C(n,5) + 3900*C(n,6) + 2940*C(n,7) + 840*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.

%F G.f.: x^2 * (1+x) * (1+56*x+306*x^2+56*x^3+x^4) / (1-x)^9.

%F a(n) = A000543(n) - A128766(n) = (A000543(n) - A337897(n)) / 2 = A128766(n) - A337897(n).

%e For a(2)=1, centering the octahedron (cube) at the origin and aligning the diagonals (edges) with the axes, color the faces (vertices) in the octants ---, --+, -++, and +++ with one color and the other 4 elements with the other color.

%t Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n,30}]

%Y Cf. A000543 (oriented), A128766(unoriented), A337897 (achiral).

%Y Other elements: A337406 (edges), A093566(n+1) (cube faces, octahedron vertices).

%Y Other polyhedra: A000332 (simplex), A093566(n+1) (cube/octahedron).

%Y Row 3 of A325014 (chiral pairs of colorings of orthoplex facets or orthotope vertices).

%Y Row 3 of A337893 (chiral pairs of colorings of orthoplex faces or orthotope peaks).

%K nonn

%O 1,3

%A _Robert A. Russell_, Sep 28 2020