A338980 Number of oriented 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, 184614999414571937405905419562270, 249584763877004334779054488506782340719383629107224173, 245395425663663491880846922641400894840783985813370231599231766603156

Offset: 0

Author

Robert A. Russell, Dec 13 2020

Keywords

• nonn
• easy
• fini

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. 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 2*bp(20)/15 + bp(30)/5 + 2*bp(40)/15 + bp(50)/6 + 13*bp(60)/150 + bp(100)/180 + bp(104)/18 + 13*bp(120)/150 + bp(150)/120 + bp(200)/180 + bp(208)/18 + bp(300)/7200 + bp(302)/16 + bp(600)/7200.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is 2*bp(24)/15 + bp(36)/5 + 2*bp(48)/15 + bp(60)/6 + 7*bp(72)/150 + bp(76)/25 + 11*bp(120)/180 + 7*bp(144)/150 + bp(152)/25 + bp(180)/120 + 11*bp(240)/180 + bp(360)/7200 + bp(364)/16 + bp(720)/7200.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is 2*bp(40)/15 + bp(60)/5 + 2*bp(80)/15 + bp(100)/6 + 13*bp(120)/150 + bp(200)/180 + bp(202)/18 + 13*bp(240)/150 + bp(300)/120 + bp(400)/180 + bp(404)/18 + bp(600)/7200 + bp(604)/16 + bp(1200)/7200.

Links

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

Crossrefs

Cf. A338981 (unoriented), A338982 (chiral), A338983 (achiral), A338964 (up to n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338948 (24-cell).

Programs

Mathematica

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[2bp[4]/15+bp[6]/5+2bp[8]/15+bp[10]/6+7bp[12]/150+bp[16]/25+bp[20]/180+bp[22]/18+7bp[24]/150+bp[30]/120+bp[32]/25+bp[40]/180+bp[44]/18+bp[60]/7200+bp[62]/16+bp[120]/7200,x]

Formula

A338964(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).

a(n) = A338981(n) + A338982(n) = 2*A338981(n) - A338983(n) = 2*A338982(n) + A338983(n).

G.f.: 2*bp(4)/15 + bp(6)/5 + 2*bp(8)/15 + bp(10)/6 + 7*bp(12)/150 + bp(16)/25 + bp(20)/180 + bp(22)/18 + 7*bp(24)/150 + bp(30)/120 + bp(32)/25 + bp(40)/180 + bp(44)/18 + bp(60)/7200 + bp(62)/16 + bp(120)/7200, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.