1,2

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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).

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).

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

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).

Sequence in context: A110847 A160472 A028512 * A088845 A013685 A108157

Adjacent sequences: A337949 A337950 A337951 * A337953 A337954 A337955

nonn,easy

Robert A. Russell, Oct 03 2020

More terms from Andrew Howroyd, Oct 22 2025

approved