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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A338964 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 subsets of a set of n colors.

Original entry on oeis.org

1, 184614999414571937405905419562272, 249584763877004334779608333505026056531601345365910986, 245395425663664490219902430658740012166428009430164733569180712873472
Offset: 1

Views

Author

Robert A. Russell, Dec 04 2020

Keywords

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. There are 7200 elements in the rotation group of the 120-cell. They divide into 41 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^120 400 x_2^3x_6^19
450 x_1^4x_2^58 20+20 x_6^20
1 x_2^60 144+144 x_2^5x_10^11
400 x_1^6x_3^38 4*12+2*144 x_10^12
20+20 x_3^40 600+600 x_12^10
144+144 x_1^10x_5^22 4*240 x_15^8
30+30 x_4^30 4*360 x_20^6
4*12+2*144 x_5^24 4*240 x_30^4
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 formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the cycle indices are:
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^600 400 x_2^6x_6^98
450 x_1^4x_2^298 20+20 x_6^100
1 x_2^300 4*12+4*144 x_10^60
400 x_1^12x_3^196 600+600 x_12^50
20+20 x_3^200 4*240 x_15^40
30+30 x_4^150 4*360 x_20^30
4*12+4*144 x_5^120 4*240 x_30^20
The formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 624*n^60 + 40*n^100 + 400*n^104 + 624*n^120 + 60*n^150 + 40*n^200 + 400*n^208 + n^300 + 450*n^302 + n^600) / 7200.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^720 2*20+400 x_6^120
450 x_1^8x_2^356 144+144 x_2^5x_10^71
1 x_2^360 4*12+2*144 x_10^72
2*20+400 x_3^240 600+600 x_12^60
30+30 x_4^180 4*240 x_15^48
144+144 x_1^10x_5^142 4*360 x_20^36
4*12+2*144 x_5^144 4*240 x_30^24
The formula is (960*n^24 + 1440*n^36 + 960*n^48 + 1200*n^60 + 336*n^72 + 288*n^76 + 440*n^120 + 336*n^144 + 288*n^152 + 60*n^180 + 440*n^240 + n^360 + 450*n^364 + n^720) / 7200.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^1200 400 x_2^3x_6^199
450 x_1^8x_2^596 20+20 x_6^200
1 x_2^600 4*12+4*144 x_10^120
400 x_1^6x_3^398 600+600 x_12^100
20+20 x_3^400 4*240 x_15^80
30+30 x_4^300 4*360 x_20^60
4*12+4*144 x_5^240 4*240 x_30^40
The formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 624*n^120 + 40*n^200 + 400*n^202 + 624*n^240 + 60*n^300 + 40*n^400 + 400*n^404 + n^600 + 450*n^604 + n^1200) / 7200.

Links

Crossrefs

Cf. A338965 (unoriented), A338966 (chiral), A338967 (achiral), A338980 (exactly n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956(16-cell vertices, 8-cell facets), A338948 (24-cell).

Programs

  • Mathematica
    Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+40n^20+400n^22+336n^24+60n^30+288n^32+40n^40+400n^44 +n^60+450n^62 +n^120)/7200,{n,10}]
  • PARI
    a(n)=(960*n^4+1440*n^6+960*n^8+1200*n^10+336*n^12+288*n^16+40*n^20+400*n^22+336*n^24+60*n^30+288*n^32+40*n^40+400*n^44+n^60+450*n^62+n^120)/7200 \\ Charles R Greathouse IV, Jul 05 2024

Formula

a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 40*n^20 + 400*n^22 + 336*n^24 + 60* n^30 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 450*n^62 + n^120) / 7200.
a(n) = Sum_{j=1..Min(n,120)} A338980(n) * binomial(n,j).
a(n) = A338965(n) + A338966(n) = 2*A338965(n) - A338967(n) = 2*A338966(n) + A338967(n).

A338981 Number of unoriented 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.

Original entry on oeis.org

0, 1, 92307499707443390526727850063502, 124792381938502167392061689732085833655832902312754962, 122697712831831745940423467267565845711242845618544066030140191642464
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

Comments

Each chiral pair is counted as one when enumerating unoriented 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 bp(20)/15 + bp(30)/10 + bp(40)/15 + bp(50)/12 + 43*bp(60)/300 + bp(66)/10 + bp(100)/360 + bp(104)/9 + bp(114)/12 + 13*bp(120)/300 + bp(150)/240 + bp(152)/8 + bp(200)/360 + bp(208)/36 + 61*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 + 3*bp(76)/25 + bp(84)/10 + 41*bp(120)/360 + bp(132)/12 + 7*bp(144)/300 + bp(152)/50 + bp(180)/240 + bp(182)/8 + 11*bp(240)/360 + 61*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 + 43*bp(120)/300 + bp(128)/10 + bp(200)/360 + bp(202)/9 + bp(216)/12 + 13*bp(240)/300 + bp(300)/240 + bp(302)/8 + bp(400)/360 + bp(404)/36 + 61*bp(600)/14400 + bp(604)/32 + bp(640)/240 + bp(1200)/14400.

Links

Crossrefs

Cf. A338980 (oriented), A338982 (chiral), A338983 (achiral), A338965 (up to n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (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

A338965(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338980(n) - A338982(n) = (A338980(n) + A338983(n)) / 2 = A338982(n) + A338983(n).
G.f.: 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, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

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.

Original entry on oeis.org

0, 0, 92307499707128546879177569498768, 124792381938502167386992798774696507063550726794469211, 122697712831831745940423455373835049129541140194826165569091574960692
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

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

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.

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.

Original entry on oeis.org

0, 1, 314843647550280564734, 5068890957389326592282175518285751, 11893730796581701705423717900461048616681772, 220581437248293418784474364671733389683204494492535
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

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

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.

A338956 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

1, 137548893254081168086800766, 11046328890861010626464488614428032600986342, 10897746068335468788318134977474134922662053604436974448, 21912802868317153141871319582922663027477920477404414535105616050
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Links

Crossrefs

Cf. A338957 (unoriented), A338958 (chiral), A338959 (achiral), A338952 (up to n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338980 (120-cell, 600-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (* binomial series *)
Drop[CoefficientList[bp[8]/6+bp[12]/4+bp[16]/12+bp[18]/18+7bp[24]/48+bp[32]/12+bp[36]/18+19bp[48]/576+bp[50]/8+bp[96]/576,x],1]

Formula

A338952(n) = Sum_{j=1..Min(n,96)} a(n) * binomial(n,j).
a(n) = A338957(n) + A338958(n) = 2*A338957(n) - A338959(n) = 2*A338958(n) + A338959(n).
Showing 1-5 of 5 results.

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