A337952 Number of oriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

1, 496, 230076, 22456756, 795467350, 14697611496, 173107727191, 1466088119056, 9651378868011, 52083991149400, 239323201136866, 962942859342036, 3465720389989936, 11343525530430016, 34210497067620525

Offset: 1

Author

Robert A. Russell, Oct 03 2020

Keywords

• nonn
• easy

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. 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 rotation group of the tesseract. 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 cycle indices for each rotation by partition. 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 Even Cycle Indices

4 6 8x_8^2

31 8 4x_1^4x_3^4 + 4x_2^2x_6^2

22 3 4x_1^4x_2^6 + 4x_4^4

211 6 4x_2^8 + 4x_4^4

1111 1 x_1^16 + 7x_2^8

Links

• Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Crossrefs

Cf. A128767 (unoriented), A337954 (chiral), A337955 (achiral).

Other elements: A331358 (tesseract edges, hyperoctahedron faces), A331354 (tesseract faces, hyperoctahedron edges), A337956 (tesseract facets, hyperoctahedron vertices).

Other polychora: A337895 (4-simplex facets/vertices), A338948 (24-cell), A338964 (120-cell, 600-cell).

Row 4 of A325012 (orthoplex facets, orthotope vertices).

Programs

Mathematica

Table[(n^16+12n^10+63n^8+68n^4+48n^2)/192,{n,30}]

Formula

a(n) = n^2 * (n^14 + 12*n^8 + 63*n^6 + 68*n^2 + 48) / 192.

a(n) = 1*C(n,1) + 494*C(n,2) + 228591*C(n,3) + 21539424*C(n,4) + 685479375*C(n,5) + 10257064650*C(n,6) + 86151316860*C(n,7) + 449772354360*C(n,8) + 1551283253100*C(n,9) + 3661969537800*C(n,10) + 6015983173200*C(n,11) + 6878457986400*C(n,12) + 5371454088000*C(n,13) + 2733402672000*C(n,14) + 817296480000*C(n,15) + 108972864000*C(n,16), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.

a(n) = A128767(n) + A337954(n) = 2*A128767(n) - A337955(n) = 2*A337954(n) + A337955(n).