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Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.
9

%I #19 Mar 08 2024 11:56:21

%S 1,21,201,1076,4025,11901,29841,66256,134001,251725,445401,750036,

%T 1211561,1888901,2856225,4205376,6048481,8520741,11783401,16026900,

%U 21474201,28384301,37055921,47831376,61100625,77305501,96944121

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

%C An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.

%C There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

%C Conjugacy Class Count Odd Cycle Indices

%C Inversion 1 x_2^4

%C Vertex rotation* 8 x_2^1x_6^1 Asterisk indicates that the

%C Edge rotation* 6 x_1^4x_2^2 operation is followed by an

%C Small face rotation* 3 x_4^2 inversion.

%C Large face rotation* 6 x_2^4

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).

%F a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.

%F a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

%F a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).

%F G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.

%t Table[n^2(7+2n^2+3n^4)/12, {n,30}]

%Y Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).

%Y Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).

%Y Other polyhedra: A006003 (tetrahedron), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).

%Y Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).

%K nonn

%O 1,2

%A _Robert A. Russell_, Sep 28 2020