A337895 Number of oriented colorings of the tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors.

1, 6, 21, 56, 127, 258, 483, 848, 1413, 2254, 3465, 5160, 7475, 10570, 14631, 19872, 26537, 34902, 45277, 58008, 73479, 92114, 114379, 140784, 171885, 208286, 250641, 299656, 356091, 420762, 494543, 578368, 673233

Offset: 1

Author

Robert A. Russell, Sep 28 2020

Keywords

• nonn

Comments

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).

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.

Partition Count Even Cycle Indices

5 24 x_5^1

311 20 x_1^2x_3^1

221 15 x_1^1x_2^2

11111 1 x_1^5

Examples

			For a(2)=6, the colors are AAAAA, AAAAB, AAABB, AABBB, ABBBB, and BBBBB.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Crossrefs

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).

Row 4 of A324999 (oriented colorings of facets or vertices of an n-simplex).

Programs

Mathematica

Table[n (24 + 35 n^2 + n^4)/60, {n, 40}]

Formula

a(n) = n * (24 + 35*n^2 + n^4) / 60.

a(n) = binomial[4+n,5] + binomial[n,5].

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.

a(n) = A000389(n+4) + A000389(n) = 2*A000389(n+4) - A132366(n-1) = 2*A000389(n) + A132366(n-1).