A331350 Number of oriented colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

1, 40, 1197, 18592, 166885, 1019880, 4738153, 17962624, 58248153, 166920040, 432738229, 1032709536, 2298857821, 4822806184, 9613704465, 18329410048, 33605960689, 59516325288, 102196242685, 170682720160, 278019522837

Offset: 1

Author

Robert A. Russell, Jan 14 2020

Keywords

• nonn
• easy

Comments

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

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 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 Even Cycle Indices

5 24 x_5^2

311 20 x_1^1x_3^3

221 15 x_1^2x_2^4

11111 1 x_1^10

Links

• G. Royle, Partitions and Permutations
• Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Crossrefs

Cf. A063843 (unoriented), A331352 (chiral), A331353 (achiral).

Other polychora: A331358 (8-cell), A331354 (16-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).

Row 4 of A327083 (simplex edges and facets) and A337883 (simplex faces and peaks).

Programs

Mathematica

Table[(24n^2 + 20n^4 + 15n^6 + n^10)/60, {n, 1, 25}]

Formula

a(n) = (24*n^2 + 20*n^4 + 15*n^6 + n^10) / 60.

a(n) = C(n,1) + 38*C(n,2) + 1080*C(n,3) + 14040*C(n,4) + 85500*C(n,5) + 274104*C(n,6) + 493920*C(n,7) + 504000*C(n,8) + 272160*C(n,9) + 60480*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = A063843(n) + A331352(n) = 2*A063843(n) - A331353(n) = 2*A331352(n) + A331353(n).