A338952 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

1, 137548893254081168086800768, 11046328890861011039111168376671536861388643, 10897746068379654103881579020805286236644252743361724416

Offset: 1

Author

Robert A. Russell, Nov 17 2020

Keywords

• nonn
• easy

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the rotation group of the 24-cell. They divide into 20 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Count Even Cycle Indices Count Even Cycle Indices

1 x_1^96 6+6+36+36 x_4^24

72 x_1^4x_2^46 32 x_2^3x_6^15

1+18 x_2^48 8+8+32 x_6^16

32 x_1^6x_3^30 72+72 x_8^12

8+8+32 x_3^32 48+48 x_12^8

Links

• Robert A. Russell, Table of n, a(n) for n = 1..30
• Index entries for linear recurrences with constant coefficients, order 97.

Crossrefs

Cf. A338953 (unoriented), A338954 (chiral), A338955 (achiral), A338956 (exactly n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338964 (120-cell, 600-cell).

Programs

Mathematica

Table[(96n^8+144n^12+48n^16+32n^18+84n^24+48n^32+32n^36+19n^48+72n^50+n^96)/576,{n,15}]

Formula

a(n) = (96*n^8 + 144*n^12 + 48*n^16 + 32*n^18 + 84*n^24 + 48*n^32 + 32*n^36 + 19*n^48 + 72*n^50 + n^96) / 576.

a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).

a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).