A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360

Offset: 1

Author

Robert A. Russell, Sep 28 2020

Keywords

• nonn
• easy

Comments

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.

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 face (octahedron vertex) 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_2^3

Vertex rotation* 8 x_6^1 Asterisk indicates that the

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

Small face rotation* 6 x_2^1x_4^1 inversion.

Large face rotation* 3 x_1^4x_2^1

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).

Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).

Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).

Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).

Programs

Mathematica

Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,55,200,560,1316},40] (* Harvey P. Dale, Feb 15 2022 *)

PARI

a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

Formula

a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.

a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).

G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 30 2020