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A337960 Number of achiral colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors. 7
1, 1048, 133875, 4211872, 61198135, 545203800, 3465030541, 17197766272, 70665499413, 250166670040, 785039389519, 2230057075104, 5826818931739, 14178299017624, 32446195329465, 70387069393408, 145689159233737 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

An achiral coloring is identical to its reflection. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

There are 60 elements in the automorphism group of the regular dodecahedron/icosahedron that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the 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_2^10

  Edge rotation*        15     x_1^4x_2^8      Asterisk indicates that the

  Vertex rotation*      20     x_2^1x_6^3      operation is followed by an

  Small face rotation*  12     x_10^2          inversion.

  Large face rotation*  12     x_10^2

LINKS

Table of n, a(n) for n=1..17.

FORMULA

a(n) = n^2 * (15*n^10 + n^8 + 20*n^2 + 24) / 60.

a(n) = 1*C(n,1) + 1046*C(n,2) + 130734*C(n,3) + 3682656*C(n,4) + 41467050*C(n,5) + 238531284*C(n,6) + 791012880*C(n,7) + 1603496160*C(n,8) + 2021060160*C(n,9) + 1546836480*C(n,10) + 658627200*C(n,11) + 119750400*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A252704(n) - A054472(n) = A054472(n) - 2*A337959(n) = A252704(n) - A337959(n).

MATHEMATICA

Table[(15n^12+n^10+20n^4+24n^2)/60, {n, 30}]

CROSSREFS

Cf. A054472 (oriented), A252704 (unoriented), A337959 (chiral).

Other elements: A337953 (edges), A337962 (dodecahedron faces, icosahedron vertices).

Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).

Sequence in context: A236273 A251375 A281033 * A331346 A164771 A031600

Adjacent sequences:  A337957 A337958 A337959 * A337961 A337962 A337963

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Oct 03 2020

STATUS

approved

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Last modified March 2 09:23 EST 2021. Contains 341746 sequences. (Running on oeis4.)