A331353 - OEIS

%I #21 Dec 28 2020 15:49:39

%S 1,28,387,2784,13125,46836,137543,349952,797769,1667500,3248971,

%T 5973408,10459917,17571204,28479375,44742656,68393873,102041532,

%U 148984339,213340000,300189141,415735188,567481047,764423424

%N Number of achiral colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

%C A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. An achiral coloring is identical to its reflection,

%C There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group.  Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy group 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  Odd Cycle Indices

%C   41         30     x_2^1x_4^2

%C   32         20     x_1^1x_3^1x_6^1

%C   2111       10     x_1^4x_2^3

%H Colin Barker, <a href="/A331353/b331353.txt">Table of n, a(n) for n = 1..1000</a>

%H G. Royle, <a href="http://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf">Partitions and Permutations</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = (5*n^3 + n^7) / 6.

%F a(n) = C(n,1) + 26*C(n,2) + 306*C(n,3) + 1400*C(n,4) + 2800*C(n,5) + 2520*C(n,6) + 840*C(n,7), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

%F a(n) = 2*A063843(n) - A331350(n) = A331350(n) - 2*A331352(n) = A063843(n) - A331352(n).

%F From _Colin Barker_, Jan 15 2020: (Start)

%F G.f.: x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8.

%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.

%F (End)

%t Table[(5 n^3 + n^7)/6, {n, 1, 25}]

%o (PARI) Vec(x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8 + O(x^25)) \\ _Colin Barker_, Jan 15 2020

%Y Cf. A331350 (oriented), A063843 (unoriented), A331352 (chiral).

%Y Other polychora: A331361 (8-cell), A331357 (16-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).

%Y Row 4 of A327086 (simplex edges and ridges) and A337886 (simplex faces and peaks).

%K nonn,easy

%O 1,2

%A _Robert A. Russell_, Jan 14 2020