OFFSET
1,2
COMMENTS
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.
LINKS
Robert A. Russell, Table of n, a(n) for n = 1..30
FORMULA
a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 40*n^20 + 400*n^22 + 336*n^24 + 60* n^30 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 450*n^62 + n^120) / 7200.
a(n) = Sum_{j=1..Min(n,120)} A338980(n) * binomial(n,j).
MATHEMATICA
Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+40n^20+400n^22+336n^24+60n^30+288n^32+40n^40+400n^44 +n^60+450n^62 +n^120)/7200, {n, 10}]
PROG
(PARI) a(n)=(960*n^4+1440*n^6+960*n^8+1200*n^10+336*n^12+288*n^16+40*n^20+400*n^22+336*n^24+60*n^30+288*n^32+40*n^40+400*n^44+n^60+450*n^62+n^120)/7200 \\ Charles R Greathouse IV, Jul 05 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Dec 04 2020
STATUS
approved