

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.


8



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
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OFFSET

1,2


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



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.
G.f.: x * (x + 4*x^2 + 10*x^3) / (1x)^6.
a(n) = 6*a(n1)  15*a(n2) + 20*a(n3)  15*a(n4) + 6*a(n5)  a(n6).  Wesley Ivan Hurt, Sep 30 2020


MATHEMATICA

Table[n(1+n)(2+n)(43n+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 *)


PROG



CROSSREFS

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).


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



