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Each chiral pair is counted as two when enumerating oriented arrangements. 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 rotation group of the 120-cell. They divide into 41 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 Even Cycle Indices Count Even Cycle Indices

1 x_1^120 400 x_2^3x_6^19

450 x_1^4x_2^58 20+20 x_6^20

1 x_2^60 144+144 x_2^5x_10^11

400 x_1^6x_3^38 4*12+2*144 x_10^12

20+20 x_3^40 600+600 x_12^10

144+144 x_1^10x_5^22 4*240 x_15^8

30+30 x_4^30 4*360 x_20^6

4*12+2*144 x_5^24 4*240 x_30^4

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 Even Cycle Indices Count Even Cycle Indices

1 x_1^600 400 x_2^6x_6^98

450 x_1^4x_2^298 20+20 x_6^100

1 x_2^300 4*12+4*144 x_10^60

400 x_1^12x_3^196 600+600 x_12^50

20+20 x_3^200 4*240 x_15^40

30+30 x_4^150 4*360 x_20^30

4*12+4*144 x_5^120 4*240 x_30^20

The formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 624*n^60 + 40*n^100 + 400*n^104 + 624*n^120 + 60*n^150 + 40*n^200 + 400*n^208 + n^300 + 450*n^302 + n^600) / 7200.

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

Count Even Cycle Indices Count Even Cycle Indices

1 x_1^720 2*20+400 x_6^120

450 x_1^8x_2^356 144+144 x_2^5x_10^71

1 x_2^360 4*12+2*144 x_10^72

2*20+400 x_3^240 600+600 x_12^60

30+30 x_4^180 4*240 x_15^48

144+144 x_1^10x_5^142 4*360 x_20^36

4*12+2*144 x_5^144 4*240 x_30^24

The formula is (960*n^24 + 1440*n^36 + 960*n^48 + 1200*n^60 + 336*n^72 + 288*n^76 + 440*n^120 + 336*n^144 + 288*n^152 + 60*n^180 + 440*n^240 + n^360 + 450*n^364 + n^720) / 7200.

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

Count Even Cycle Indices Count Even Cycle Indices

1 x_1^1200 400 x_2^3x_6^199

450 x_1^8x_2^596 20+20 x_6^200

1 x_2^600 4*12+4*144 x_10^120

400 x_1^6x_3^398 600+600 x_12^100

20+20 x_3^400 4*240 x_15^80

30+30 x_4^300 4*360 x_20^60

4*12+4*144 x_5^240 4*240 x_30^40

The formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 624*n^120 + 40*n^200 + 400*n^202 + 624*n^240 + 60*n^300 + 40*n^400 + 400*n^404 + n^600 + 450*n^604 + n^1200) / 7200.

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.

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 formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 2064*n^60 + 1440*n^66 + 40*n^100 + 1600*n^104 + 1200*n^114 + 624*n^120 + 60*n^150 + 1800*n^152 + 40*n^200 + 400*n^208 + 61*n^300 + 450*n^302 + 60*n^330 + n^600) / 14400.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 + 1728 n^76 + 1440 n^84 + 1640 n^120 + 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 + 1800 n^182 + 440 n^240 + 61 n^360 + 450 n^364 + 60 n^396 + n^720) / 14400.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 2064*n^120 + 1440*n^128 + 40*n^200 + 1600*n^202 + 1200*n^216 + 624*n^240 + 60*n^300 + 1800*n^302 + 40*n^400 + 400*n^404 + 61*n^600 + 450*n^604 + 60*n^640 + n^1200) / 14400.

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.

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 formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 - 816*n^60 - 1440*n^66 + 40*n^100 - 800*n^104 - 1200*n^114 + 624*n^120 + 60*n^150 - 1800*n^152 + 40*n^200 + 400*n^208 - 59*n^300 + 450*n^302 - 60*n^330 + n^600) / 14400.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 - 1152 n^76 - 1440 n^84 - 760 n^120 - 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 - 1800 n^182 + 440 n^240 - 59 n^360 + 450 n^364 - 60 n^396 + n^720) / 14400.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 - 816*n^120 - 1440*n^128 + 40*n^200 - 800*n^202 - 1200*n^216 + 624*n^240 + 60*n^300 - 1800*n^302 + 40*n^400 + 400*n^404 - 59*n^600 + 450*n^604 - 60*n^640 + n^1200) / 14400.

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)

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. An achiral coloring is identical to its reflection,

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 group of the permutation. The first formula 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_2^1x_4^2

32 20 x_1^1x_3^1x_6^1

2111 10 x_1^4x_2^3

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the square faces of a tesseract {4,3,3} with n available colors.

There are 192 elements in the automorphism group of the 4-dimensional orthoplex that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula 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

4 6 8x_2^2x_4^5

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

22 3 8x_1^2x_2^1x_4^5

211 6 2x_1^2x_2^11 + 2x_1^6x_2^9 + 4x_2^2x_4^5

1111 1 4x_1^12x_2^6 + 4x_2^12

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.

There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula 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

4 6 8x_4^8

31 8 4x_1^2x_3^2x_6^4 + 4x_2^1x_6^5

22 3 8x_4^8

211 6 2x_1^8x_2^12 + 2x_2^16 + 4x_4^8

1111 1 4x_1^8x_2^12 + 4x_2^16

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

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

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. There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. 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 hyperoctahedron facet (tesseract vertex) cycle indices for each member of such a class. 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 Odd Cycle Indices

4 6 8x_1^2x_2^1x_4^3

31 8 8x_2^2x_6^2

22 3 8x_4^4

211 6 2x_1^8x_2^4 + 2x_2^8 + 4x_4^4

1111 1 8x_2^8