%I #13 Mar 13 2024 13:48:35
%S 1,137548893254081168086800768,
%T 11046328890861011039111168376671536861388643,
%U 10897746068379654103881579020805286236644252743361724416
%N Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.
%C Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the rotation group of the 24-cell. They divide into 20 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C Count Even Cycle Indices Count Even Cycle Indices
%C 1 x_1^96 6+6+36+36 x_4^24
%C 72 x_1^4x_2^46 32 x_2^3x_6^15
%C 1+18 x_2^48 8+8+32 x_6^16
%C 32 x_1^6x_3^30 72+72 x_8^12
%C 8+8+32 x_3^32 48+48 x_12^8
%H Robert A. Russell, <a href="/A338952/b338952.txt">Table of n, a(n) for n = 1..30</a>
%H <a href="/index/Rec#order_97">Index entries for linear recurrences with constant coefficients</a>, order 97.
%F a(n) = (96*n^8 + 144*n^12 + 48*n^16 + 32*n^18 + 84*n^24 + 48*n^32 + 32*n^36 + 19*n^48 + 72*n^50 + n^96) / 576.
%F a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).
%F a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).
%t Table[(96n^8+144n^12+48n^16+32n^18+84n^24+48n^32+32n^36+19n^48+72n^50+n^96)/576,{n,15}]
%Y Cf. A338953 (unoriented), A338954 (chiral), A338955 (achiral), A338956 (exactly n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338964 (120-cell, 600-cell).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Nov 17 2020