A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

1, 8200, 9080559, 1503323520, 81461669375, 2146080958056, 34228350856910, 377534786525184, 3140004522270465, 20896479183085000, 116094911796177061, 555622588428635520, 2346039511676401359, 8903083257215729960

Offset: 1

Author

Robert A. Russell, Jan 14 2020

Keywords

• nonn
• easy

Comments

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the square faces of a tesseract {4,3,3} with n available colors.

There are 192 elements in the automorphism group of the 4-dimensional orthoplex 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_2^2x_4^5

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

22 3 8x_1^2x_2^1x_4^5

211 6 2x_1^2x_2^11 + 2x_1^6x_2^9 + 4x_2^2x_4^5

1111 1 4x_1^12x_2^6 + 4x_2^12

Links

• G. Royle, Partitions and Permutations
• Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Crossrefs

Cf. A331354 (oriented), A331355 (unoriented), A331356 (chiral).

Other polychora: A331353 (5-cell), A331361 (8-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).

Row 4 of A337414 (orthoplex edges, orthotope ridges) and A337890 (orthotope faces, orthoplex peaks).

Programs

Mathematica

Table[(8n^4 + 8n^6 + 18n^7 + 6n^8 + n^12 + 3n^13 + 3n^15 + n^18)/48, {n, 1, 25}]

Formula

a(n) = (8*n^4 + 8*n^6 + 18*n^7 + 6*n^8 + n^12 + 3*n^13 + 3*n^15 + n^18) / 48.

a(n) = C(n,1) + 8198*C(n,2) + 9055962*C(n,3) + 1467050480*C(n,4) + 74035775370*C(n,5) + 1679679306420*C(n,6) + 20864180531565*C(n,7) + 159341117375160*C(n,8) + 804216787965360*C(n,9) + 2808560520334800*C(n,10) + 6981656802951600*C(n,11) + 12540346820971200*C(n,12) + 16328843044113600*C(n,13) + 15272715797539200*C(n,14) + 10003790644848000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = 2*A331355(n) - A331354(n) = A331354(n) - 2*A331356(n) = A331355(n) - A331356(n).