OFFSET
1,2
COMMENTS
An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. 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 edge (or face) 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_1^24x_2^36 96 x_1^2x_2^2x_3^2x_6^14
12 x_1^8x_2^44 96 x_3^8x_6^12
12+12 x_3^48 96 x_2^3x_6^15
72+72 x_4^24 96 x_6^16
LINKS
Robert A. Russell, Table of n, a(n) for n = 1..30
FORMULA
MATHEMATICA
Table[(8n^16+8n^18+16n^20+12n^24+2n^48+n^52+n^60)/48, {n, 15}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Nov 17 2020
STATUS
approved