%I #14 Mar 11 2024 13:11:38
%S 1,6,21,56,127,258,483,848,1413,2254,3465,5160,7475,10570,14631,19872,
%T 26537,34902,45277,58008,73479,92114,114379,140784,171885,208286,
%U 250641,299656,356091,420762,494543,578368,673233
%N Number of oriented colorings of the tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors.
%C Each chiral pair is counted as two when enumerating oriented arrangements. Also called a 5-cell or pentachoron. The Schläfli symbol is {3,3,3}, and it has 5 tetrahedral facets (vertices).
%C There are 60 elements in the rotation group of the 4-dimensional simplex. Each is an even permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C Partition Count Even Cycle Indices
%C 5 24 x_5^1
%C 311 20 x_1^2x_3^1
%C 221 15 x_1^1x_2^2
%C 11111 1 x_1^5
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, -15, 20, -15, 6, -1).
%F a(n) = n * (24 + 35*n^2 + n^4) / 60.
%F a(n) = binomial[4+n,5] + binomial[n,5].
%F a(n) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 2*C(n,5), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F a(n) = A000389(n+4) + A000389(n) = 2*A000389(n+4) - A132366(n-1) = 2*A000389(n) + A132366(n-1).
%e For a(2)=6, the colors are AAAAA, AAAAB, AAABB, AABBB, ABBBB, and BBBBB.
%t Table[n (24 + 35 n^2 + n^4)/60, {n, 40}]
%Y Cf. A000389(n+4) (unoriented), A000389(chiral), A132366(n-1) (achiral), A331350 (edges, faces), A337952 (8-cell vertices, 16-cell facets), A337956(16-cell vertices, 8-cell facets), A338948 (24-cell), A338964 (120-cell, 600-cell).
%Y Row 4 of A324999 (oriented colorings of facets or vertices of an n-simplex).
%K nonn
%O 1,2
%A _Robert A. Russell_, Sep 28 2020