A337955 Number of achiral colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

1, 308, 34128, 1056576, 15303750, 136236276, 865711763, 4296782848, 17656466751, 62510672500, 196174554026, 557301826368, 1456216515468, 3543525156276, 8109415963125, 17592637669376, 36414622551373

Offset: 1

Author

Robert A. Russell, Oct 03 2020

Keywords

• nonn
• easy

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual. There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each involves a permutation of the axes that can be associated with a partition of 4 based on the conjugacy class of the permutation. This table shows the hyperoctahedron facet (tesseract vertex) cycle indices for each member of such a class. The first formula is obtained by averaging these cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Partition Count Odd Cycle Indices

4 6 8x_1^2x_2^1x_4^3

31 8 8x_2^2x_6^2

22 3 8x_4^4

211 6 2x_1^8x_2^4 + 2x_2^8 + 4x_4^4

1111 1 8x_2^8

Links

• Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Crossrefs

Cf. A337952 (oriented), A128767 (unoriented), A337954 (chiral).

Other elements: A331361 (tesseract edges, hyperoctahedron faces), A331357 (tesseract faces, hyperoctahedron edges), A337958 (tesseract facets, hyperoctahedron vertices).

Other polychora: A132366(n-1) (4-simplex facets/vertices), A338951 (24-cell), A338967 (120-cell, 600-cell).

Row 4 of A325015 (orthoplex facets, orthotope vertices).

Programs

Mathematica

Table[(3n^12+5n^8+12n^6+28n^4)/48,{n,30}]

Formula

a(n) = n^4 * (3*n^8 + 5*n^4 + 12*n^2 + 28) / 48.

a(n) = 1*C(n,1) + 306*C(n,2) + 33207*C(n,3) + 921908*C(n,4) + 10359075*C(n,5) + 59584470*C(n,6) + 197644440*C(n,7) + 400752240*C(n,8) + 505197000*C(n,9) + 386694000*C(n,10) + 164656800*C(n,11) + 29937600*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A128767(n) - A337952(n) = A337952(n) - 2*A337954(n) = A128767(n) - A337954(n).