A338983 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

0, 1, 314843647550280564734, 5068890957389326592282175518285751, 11893730796581701705423717900461048616681772, 220581437248293418784474364671733389683204494492535

Offset: 0

Author

Robert A. Russell, Dec 13 2020

Keywords

• fini
• nonn
• easy

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>75, a(n) = 0.

Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.

For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(60)/5 + bp(66)/5 + bp(104)/6 + bp(114)/6 + bp(152)/4 + bp(300)/120 + bp(330)/120.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(76)/5 + bp(84)/5 + bp(120)/6 + bp(132)/6 + bp(182)/4 + bp(360)/120 + bp(396)/120.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(120)/5 + bp(128)/5 + bp(202)/6 + bp(216)/6 + bp(302)/4 + bp(600)/120 + bp(640)/120.

Links

• Robert A. Russell, Table of n, a(n) for n = 0..75

Crossrefs

Cf. A338980 (oriented), A338981 (unoriented), A338982 (chiral), A338967 (up to n colors), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell).

Programs

Mathematica

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[17]/5+bp[19]/5+bp[23]/6+bp[27]/6+bp[31]/4+bp[61]/120+bp[75]/120,x]

Formula

A338967(n) = Sum_{j=1..Min(n,75)} a(n) * binomial(n,j).

a(n) = 2*A338981(n) - A338980(n) = A338980(n) - 2*A338982(n) = A338981(n) - A338982(n).

G.f.: bp(17)/5 + bp(19)/5 + bp(23)/6 + bp(27)/6 + bp(31)/4 + bp(61)/120 + bp(75)/120, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.