cp's OEIS Frontend

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)

Sequence is column #5 of A084546: a(n) = A084546(n+1, 4).

All elements of the sequence are multiples of 15.

a(n-1) is the number of chiral pairs of colorings of the 8 cubic facets of a tesseract (hypercube) with Schläfli symbol {4,3,3} or of the 8 vertices of a hyperoctahedron with Schläfli symbol {3,3,4}. Both figures are regular 4-D polyhedra and they are mutually dual. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

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 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

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.

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.

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.