cp's OEIS Frontend

A338952 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.

%I A338952 #13 Mar 13 2024 13:48:35
%S A338952 1,137548893254081168086800768,
%T A338952 11046328890861011039111168376671536861388643,
%U A338952 10897746068379654103881579020805286236644252743361724416
%N A338952 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 A338952 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 A338952    Count   Even Cycle Indices      Count   Even Cycle Indices
%C A338952        1   x_1^96              6+6+36+36   x_4^24
%C A338952       72   x_1^4x_2^46                32   x_2^3x_6^15
%C A338952     1+18   x_2^48                 8+8+32   x_6^16
%C A338952       32   x_1^6x_3^30             72+72   x_8^12
%C A338952   8+8+32   x_3^32                  48+48   x_12^8
%H A338952 Robert A. Russell, <a href="/A338952/b338952.txt">Table of n, a(n) for n = 1..30</a>
%H A338952 <a href="/index/Rec#order_97">Index entries for linear recurrences with constant coefficients</a>, order 97.
%F A338952 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 A338952 a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).
%F A338952 a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).
%t A338952 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 A338952 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 A338952 nonn,easy
%O A338952 1,2
%A A338952 _Robert A. Russell_, Nov 17 2020