A331351 Number of achiral colorings of the edges of a cube or regular octahedron.

1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935, 15041950, 31740841, 62830560, 117855192, 211141490, 363551700, 604679936, 975561405, 1531968822, 2348375395, 3522668800, 5181705606, 7487800650, 10646250902

Offset: 1

Author

Robert A. Russell, Jan 14 2020

Keywords

• nonn
• easy

Comments

A cube has 8 vertices and 12 edges. A regular octahedron has 6 vertices and 12 edges. An achiral coloring is identical to its reflection.

From Robert A. Russell, Oct 08 2020: (Start)

The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.

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 edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Conjugacy Class Count Odd Cycle Indices

Inversion 1 x_2^6

Vertex rotation* 8 x_6^2 Asterisk indicates that the

Edge rotation* 6 x_1^2x_2^5 operation is followed by an

Small face rotation* 3 x_4^3 inversion.

Large face rotation* 6 x_1^4x_2^4 (End)

Links

• G. Royle, Partitions and Permutations
• Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Crossrefs

Cf. A060530 (oriented), A199406 (unoriented), A337406 (chiral), A337897 (octahedron faces, cube vertices), A337898 (cube faces, octahedron vertices), A037270 (tetrahedron), A337953 (dodecahedron, icosahedron).

Row 3 of A337410 (orthotope edges, orthoplex ridges) and A337414 (orthoplex edges, orthotope ridges).

Programs

Mathematica

Table[(8n^2 + 6n^3 + n^6 + 6n^7 + 3n^8)/24, {n, 1, 30}]
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935}, 25]

Formula

a(n) = (8*n^2 + 6*n^3 + n^6 + 6*n^7 + 3*n^8) / 24.

a(n) = 1*C(n,1) + 68*C(n,2) + 1200*C(n,3) + 7268*C(n,4) + 20025*C(n,5) + 27750*C(n,6) + 18900*C(n,7) + 5040*C(n,8), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = 2*A199406(n) - A060530(n) = A060530(n) - 2*A337406(n) = A199406(n) - A337406(n). - Robert A. Russell, Oct 08 2020

G.f.: (x + 61*x^2 + 813*x^3 + 2253*x^4 + 1628*x^5 + 282*x^6 + 2*x^7) / (1-x)^9.

E.g.f.: (1/24)*exp(x)*x*(24 + 816*x + 4800*x^2 + 7268*x^3 + 4005*x^4 + 925*x^5 + 90*x^6 + 3*x^7). - Stefano Spezia, Jan 17 2020