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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. 13
1, 314843647550280564736, 5068890957390271123224826359979956, 11893730816857265534982913331475052373213184, 220581496716947452381892465686737251285705566406250 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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).
MATHEMATICA
Table[(24n^17+24n^19+20n^23+20n^27+30n^31+n^61+n^75)/120, {n, 10}]
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).
Sequence in context: A181790 A252504 A338983 * A239924 A181791 A217431
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Dec 04 2020
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)