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A338964 Number of oriented 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, 184614999414571937405905419562272, 249584763877004334779608333505026056531601345365910986, 245395425663664490219902430658740012166428009430164733569180712873472 (list; graph; refs; listen; history; text; internal format)
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).

a(n) = A338965(n) + A338966(n) = 2*A338965(n) - A338967(n) = 2*A338966(n) + A338967(n).

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}]

CROSSREFS

Cf. A338965 (unoriented), A338966 (chiral), A338967 (achiral), A338980 (exactly n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956(16-cell vertices, 8-cell facets), A338948 (24-cell).

Sequence in context: A338965 A095458 A338980 * A277140 A083104 A115531

Adjacent sequences:  A338961 A338962 A338963 * A338965 A338966 A338967

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Dec 04 2020

STATUS

approved

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Last modified June 27 04:16 EDT 2022. Contains 354888 sequences. (Running on oeis4.)