A331352 Number of chiral pairs of colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

0, 6, 405, 7904, 76880, 486522, 2300305, 8806336, 28725192, 82626270, 214744629, 513368064, 1144198952, 2402617490, 4792612545, 9142333696, 16768783408, 29707141878, 51023629173, 85234690080, 138859666848

Offset: 1

Author

Robert A. Russell, Jan 14 2020

Keywords

• nonn
• easy

Comments

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. The chiral colorings of its edges come in pairs, each the reflection of the other.

Links

• Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Crossrefs

Cf. A331350 (oriented), A063843 (unoriented), A331353 (achiral).

Other polychora: A331360 (8-cell), A331356 (16-cell), A338954 (24-cell), A338966 (120-cell, 600-cell).

Row 4 of A327085 (simplex edges and ridges) and A337885 (simplex faces and peaks).

Programs

Mathematica

Table[(24n^2 - 50n^3 + 20n^4 + 15n^6 - 10n^7 + n^10)/120, {n, 1, 25}]

Formula

a(n) = (24*n^2 - 50*n^3 + 20*n^4 + 15*n^6 - 10*n^7 + n^10) / 120.

a(n) = 6*C(n,2) + 387*C(n,3) + 6320*C(n,4) + 41350*C(n,5) + 135792*C(n,6) + 246540*C(n,7) + 252000*C(n,8) + 136080*C(n,9) + 30240*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = A331350(n) - A063843(n) = (A331350(n) - A331353(n)) / 2 = A063843(n) - A331353(n).