A331361 Number of achiral colorings of the edges of a tesseract with n available colors.

1, 93024, 294157089, 91983927296, 7960001890625, 304914963625056, 6652124939544609, 96100248309858304, 1013293206632601441, 8334166666733500000, 56066328722011832961, 319495406392484665344

Offset: 1

Author

Robert A. Russell, Jan 14 2020

Keywords

• nonn
• easy

Comments

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.

There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. 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 Odd Cycle Indices

4 6 8x_4^8

31 8 4x_1^2x_3^2x_6^4 + 4x_2^1x_6^5

22 3 8x_4^8

211 6 2x_1^8x_2^12 + 2x_2^16 + 4x_4^8

1111 1 4x_1^8x_2^12 + 4x_2^16

Links

• G. Royle, Partitions and Permutations
• Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Crossrefs

Cf. A331358 (oriented), A331359 (unoriented), A331360 (chiral).

Cf. A331353 (simplex), A331357 (orthoplex), A338955 (24-cell), A338967 (120-cell, 600-cell).

Programs

Mathematica

Table[(2n^6 + 8n^8 + n^16 + n^20)/12, {n, 1, 25}]

Formula

a(n) = (2*n^6 + 8*n^8 + n^16 + n^20) / 12.

a(n) = C(n,1) + 93022*C(n,2) + 293878020*C(n,3) + 90807857080*C(n,4) + 7503022894800*C(n,5) + 258528829444320*C(n,6) + 4681671089961600*C(n,7) + 50981530073846400*C(n,8) + 363246007692204000*C(n,9) + 1789536284820648000*C(n,10) + 6323058513173001600*C(n,11) + 16406578807069651200*C(n,12) + 31689737477798400000*C(n,13) + 45786987328642560000*C(n,14) + 49291621471572480000*C(n,15) + 38970361271761920000*C(n,16) + 21972146261345280000*C(n,17) + 8363100653107200000*C(n,18) + 1926047423139840000*C(n,19) + 202741834014720000*C(n,20), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = 2*A331359(n) - A331358(n) = A331358(n) - 2*A331360(n) = A331359(n) - A331360(n).