1,2

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,

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.

Partition Count Odd Cycle Indices

41 30 x_2^1x_4^2

32 20 x_1^1x_3^1x_6^1

2111 10 x_1^4x_2^3

Colin Barker, Table of n, a(n) for n = 1..1000

Gordon Royle, Partitions and Permutations.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

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

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.

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

From Colin Barker, Jan 15 2020: (Start)

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

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.

(End)

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

(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

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

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

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

Sequence in context: A022623 A077507 A130608 * A177108 A283637 A126921

Adjacent sequences: A331350 A331351 A331352 * A331354 A331355 A331356

nonn,easy

Robert A. Russell, Jan 14 2020

approved