A338967 Number of achiral colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.

1, 314843647550280564736, 5068890957390271123224826359979956, 11893730816857265534982913331475052373213184, 220581496716947452381892465686737251285705566406250

Offset: 1

Author

Robert A. Russell, Dec 04 2020

Keywords

• nonn
• easy

Comments

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.

Links

• Robert A. Russell, Table of n, a(n) for n = 1..30
• Index entries for linear recurrences with constant coefficients, order 76.

Crossrefs

Cf. A338964 (oriented), A338965 (unoriented), A338966 (chiral), A338983 (exactly n colors), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958(16-cell vertices, 8-cell facets), A338951 (24-cell).

Programs

Mathematica

Table[(24n^17+24n^19+20n^23+20n^27+30n^31+n^61+n^75)/120,{n,10}]

PARI

a(n)=(24*n^17+24*n^19+20*n^23+20*n^27+30*n^31+n^61+n^75)/120 \\ Charles R Greathouse IV, Jul 05 2024

Formula

a(n) = (24*n^17 + 24*n^19 + 20*n^23 + 20*n^27 + 30*n^31 + n^61 + n^75) / 120.

a(n) = Sum_{j=1..Min(n,75)} A338983(n) * binomial(n,j).

a(n) = 2*A338965(n) - A338964(n) =(A338964(n) - 2*A338966(n)) / 2 = A338965(n) - A338966(n).