A338951 Number of achiral colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

1, 6504, 8416440, 1455789440, 80139247500, 2125945744776, 34026498820524, 376045864704000, 3131319814422255, 20854395850585000, 115919421344402676, 554976171149122944, 2343894146343268610, 8896568181794053320

Offset: 1

Author

Robert A. Russell, Nov 17 2020

Keywords

• nonn
• easy

Comments

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Count Odd Cycle Indices Count Odd Cycle Indices

12 x_1^12x_2^6 72 x_2^2x_4^5

12 x_1^6x_2^9 96 x_1^2x_2^2x_6^3

12 x_1^2x_2^11 96 x_2^3x_3^2x_6^2

12 x_2^12 96 x_3^4x_6^2

72 x_1^2x_2^1x_4^5 96 x_6^4

Links

• Robert A. Russell, Table of n, a(n) for n = 1..30
• Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Crossrefs

Cf. A338948 (oriented), A338949 (unoriented), A338950 (chiral), A338955 (edges, faces), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338967 (120-cell, 600-cell).

Programs

Mathematica

Table[(8n^4+8n^6+22n^7+6n^8+n^12+n^13+n^15+n^18)/48,{n,15}]

Formula

a(n) = (8*n^4 + 8*n^6 + 22*n^7 + 6*n^8 + n^12 + n^13 + n^15 + n^18) / 48.

a(n) = 1*C(n,1) + 6502*C(n,2) + 8396931*C(n,3) + 1422162700*C(n,4) + 72944399665*C(n,5) + 1666778870130*C(n,6) + 20777144613015*C(n,7) + 158973991255800*C(n,8) + 803196369526320*C(n,9) + 2806639981714800*C(n,10) + 6979192091902800*C(n,11) + 12538220293368000*C(n,12) + 16327662245294400*C(n,13) + 15272334392515200*C(n,14) + 10003736158416000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A338949(n) - A338948(n) = A338948(n) - 2*A338950(n) = A338949(n) - A338950(n).