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An achiral coloring is identical to its reflection. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. There are 7200 elements in the automorphism group of the 120-cell that are not in its rotation group. They divide into 9 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

60 x_1^30x_2^45 1200 x_1^2x_2^2x_6^19

60 x_1^2x_2^59 720+720 x_2^5x_5^6x_10^8

1800 x_2^2x_4^29 720+720 x_1^2x_2^4x_10^11

1200 x_2^3x_3^10x_6^14

Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.

For the 600 facets of the 600-cell (vertices of the 120-cell), the cycle indices are:

Count Odd Cycle Indices Count Odd Cycle Indices

60 x_1^60x_2^270 1200 x_2^6x_6^98

60 x_2^300 720+720 x_5^12x_10^54

1800 x_1^2x_2^1x_4^149 720+720 x_10^60

1200 x_2^6x_3^20x_6^88

The formula is (24*n^60 + 24*n^66 + 20*n^104 + 20*n^114 + 30*n^152 + n^300 + n^330) / 120.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:

Count Odd Cycle Indices Count Odd Cycle Indices

60 x_1^72x_2^324 1200 x_6^120

60 x_2^360 720+720 x_1^2x_2^4x_5^14x_10^64

1800 x_2^4x_4^178 720+720 x_2^5x_10^71

1200 x_3^24x_6^108

The formula is (24*n^76 + 24*n^84 + 20*n^120 + 20*n^132 + 30*n^182 + n^360 + n^396) / 120.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:

Count Odd Cycle Indices Count Odd Cycle Indices

60 x_1^80x_2^560 1200 x_2^3x_6^199

60 x_2^600 720+720 x_5^16x_10^112

1800 x_2^4x_4^298 720+720 x_10^120

1200 x_1^2x_2^2x_3^26x_6^186

The formula is (24*n^120 + 24*n^128 + 20*n^202 + 20*n^216 + 30*n^302 + n^600 + n^640) / 120.

From Robert A. Russell, Oct 09 2020: (Start)

a(n-1) is the number of achiral colorings of the 5 tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors. An achiral arrangement is identical to its reflection. The 4-dimensional simplex is also called a 5-cell or pentachoron. Its Schläfli symbol is {3,3,3}.

There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group. Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula for a(n-1) is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Partition Count Odd Cycle Indices

41 30 x_1x_4^1

32 20 x_2^1x_3^1

2111 10 x_1^3x_2^1 (End)

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

I assume this refers to colorings of the vertices of the cube. - N. J. A. Sloane, Apr 06 2007

Number of unoriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract (4-D cube) using up to n colors. Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual. - Robert A. Russell, Oct 03 2020

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual. There are 192 elements in the rotation group of the tesseract. Each involves a permutation of the axes that can be associated with a partition of 4 based on the conjugacy class of the permutation. This table shows the cycle indices for each rotation by partition. The first formula is obtained by averaging these cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Partition Count Even Cycle Indices

4 6 8x_8^2

31 8 4x_1^4x_3^4 + 4x_2^2x_6^2

22 3 4x_1^4x_2^6 + 4x_4^4

211 6 4x_2^8 + 4x_4^4

1111 1 x_1^16 + 7x_2^8

Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.

An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.

An achiral coloring is identical to its reflection. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>75, a(n) = 0.

Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.

For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(60)/5 + bp(66)/5 + bp(104)/6 + bp(114)/6 + bp(152)/4 + bp(300)/120 + bp(330)/120.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(76)/5 + bp(84)/5 + bp(120)/6 + bp(132)/6 + bp(182)/4 + bp(360)/120 + bp(396)/120.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(120)/5 + bp(128)/5 + bp(202)/6 + bp(216)/6 + bp(302)/4 + bp(600)/120 + bp(640)/120.