A338982 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, 0, 92307499707128546879177569498768, 124792381938502167386992798774696507063550726794469211, 122697712831831745940423455373835049129541140194826165569091574960692

Offset: 0

Author

Robert A. Russell, Dec 13 2020

Keywords

• fini
• nonn
• easy

Comments

Each member of a chiral pair is a reflection but not a rotation of the other. 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>120, 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(20)/15 + bp(30)/10 + bp(40)/15 + bp(50)/12 - 17*bp(60)/300 - bp(66)/10 + bp(100)/360 - bp(104)/18 - bp(114)/12 + 13*bp(120)/300 + bp(150)/240 - bp(152)/8 + bp(200)/360 + bp(208)/36 - 59*bp(300)/14400 + bp(302)/32 - bp(330)/240 + bp(600)/14400.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(24)/15 + bp(36)/10 + bp(48)/15 + bp(60)/12 + 7*bp(72)/300 - 2*bp(76)/25 - bp(84)/10 - 19*bp(120)/360 - bp(132)/12 + 7*bp(144)/300 + bp(152)/50 + bp(180)/240 - bp(182)/8 + 11*bp(240)/360 - 59*bp(360)/14400 + bp(364)/32 - bp(396)/240 + bp(720)/14400.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(40)/15 + bp(60)/10 + bp(80)/15 + bp(100)/12 - 17*bp(120)/300 - bp(128)/10 + bp(200)/360 - bp(202)/18 - bp(216)/12 + 13*bp(240)/300 + bp(300)/240 - bp(302)/8 + bp(400)/360 + bp(404)/36 - 59*bp(600)/14400 + bp(604)/32 - bp(640)/240 + bp(1200)/14400.

Links

• Robert A. Russell, Table of n, a(n) for n = 0..120

Crossrefs

Cf. A338980 (oriented), A338981 (unoriented), A338983 (achiral), A338966 (up to n colors), A000389 (5-cell), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell).

Programs

Mathematica

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[4]/15+bp[6]/10+bp[8]/15+bp[10]/12+7bp[12]/300+bp[16]/50-bp[17]/10-bp[19]/10+bp[20]/360+bp[22]/36-bp[23]/12+7bp[24]/300-bp[27]/12+bp[30]/240-bp[31]/8+bp[32]/50+bp[40]/360+bp[44]/36+bp[60]/14400-bp[61]/240+bp[62]/32-bp[75]/240+bp[120]/14400,x]

Formula

A338966(n) = Sum_{j=2..Min(n,120)} a(n) * binomial(n,j).

a(n) = A338980(n) - A338981(n) = (A338980(n) - A338983(n)) / 2 = A338981(n) - A338983(n).

G.f.: bp(4)/15 + bp(6)/10 + bp(8)/15 + bp(10)/12 + 7*bp(12)/300 + bp(16)/50 - bp(17)/10 - bp(19)/10 + bp(20)/360 + bp(22)/36 - bp(23)/12 + 7*bp(24)/300 - bp(27)/12 + bp(30)/240 - bp(31)/8 + bp(32)/50 + bp(40)/360 + bp(44)/36 + bp(60)/14400 - bp(61)/240 + bp(62)/32 - bp(75)/240 + bp(120)/14400, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.