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%I A338964 #21 Jul 05 2024 14:09:49
%S A338964 1,184614999414571937405905419562272,
%T A338964 249584763877004334779608333505026056531601345365910986,
%U A338964 245395425663664490219902430658740012166428009430164733569180712873472
%N 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.
%C A338964 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.
%C A338964 Count Even Cycle Indices Count Even Cycle Indices
%C A338964 1 x_1^120 400 x_2^3x_6^19
%C A338964 450 x_1^4x_2^58 20+20 x_6^20
%C A338964 1 x_2^60 144+144 x_2^5x_10^11
%C A338964 400 x_1^6x_3^38 4*12+2*144 x_10^12
%C A338964 20+20 x_3^40 600+600 x_12^10
%C A338964 144+144 x_1^10x_5^22 4*240 x_15^8
%C A338964 30+30 x_4^30 4*360 x_20^6
%C A338964 4*12+2*144 x_5^24 4*240 x_30^4
%C A338964 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.
%C A338964 For the 600 facets of the 600-cell (vertices of the 120-cell), the cycle indices are:
%C A338964 Count Even Cycle Indices Count Even Cycle Indices
%C A338964 1 x_1^600 400 x_2^6x_6^98
%C A338964 450 x_1^4x_2^298 20+20 x_6^100
%C A338964 1 x_2^300 4*12+4*144 x_10^60
%C A338964 400 x_1^12x_3^196 600+600 x_12^50
%C A338964 20+20 x_3^200 4*240 x_15^40
%C A338964 30+30 x_4^150 4*360 x_20^30
%C A338964 4*12+4*144 x_5^120 4*240 x_30^20
%C A338964 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.
%C A338964 For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:
%C A338964 Count Even Cycle Indices Count Even Cycle Indices
%C A338964 1 x_1^720 2*20+400 x_6^120
%C A338964 450 x_1^8x_2^356 144+144 x_2^5x_10^71
%C A338964 1 x_2^360 4*12+2*144 x_10^72
%C A338964 2*20+400 x_3^240 600+600 x_12^60
%C A338964 30+30 x_4^180 4*240 x_15^48
%C A338964 144+144 x_1^10x_5^142 4*360 x_20^36
%C A338964 4*12+2*144 x_5^144 4*240 x_30^24
%C A338964 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.
%C A338964 For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:
%C A338964 Count Even Cycle Indices Count Even Cycle Indices
%C A338964 1 x_1^1200 400 x_2^3x_6^199
%C A338964 450 x_1^8x_2^596 20+20 x_6^200
%C A338964 1 x_2^600 4*12+4*144 x_10^120
%C A338964 400 x_1^6x_3^398 600+600 x_12^100
%C A338964 20+20 x_3^400 4*240 x_15^80
%C A338964 30+30 x_4^300 4*360 x_20^60
%C A338964 4*12+4*144 x_5^240 4*240 x_30^40
%C A338964 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.
%H A338964 Robert A. Russell, <a href="/A338964/b338964.txt">Table of n, a(n) for n = 1..30</a>
%H A338964 <a href="/index/Rec#order_121">Index entries for linear recurrences with constant coefficients</a>, order 121.
%F A338964 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.
%F A338964 a(n) = Sum_{j=1..Min(n,120)} A338980(n) * binomial(n,j).
%F A338964 a(n) = A338965(n) + A338966(n) = 2*A338965(n) - A338967(n) = 2*A338966(n) + A338967(n).
%t A338964 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}]
%o A338964 (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
%Y A338964 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).
%K A338964 nonn,easy
%O A338964 1,2
%A A338964 _Robert A. Russell_, Dec 04 2020