A338955 -id:A338955 - cp's OEIS frontend

A331353 Number of achiral colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

1, 28, 387, 2784, 13125, 46836, 137543, 349952, 797769, 1667500, 3248971, 5973408, 10459917, 17571204, 28479375, 44742656, 68393873, 102041532, 148984339, 213340000, 300189141, 415735188, 567481047, 764423424

Offset: 1

Robert A. Russell, Jan 14 2020

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. An achiral coloring is identical to its reflection,
There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group.  Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy group of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  41         30     x_2^1x_4^2
  32         20     x_1^1x_3^1x_6^1
  2111       10     x_1^4x_2^3

Colin Barker, Table of n, a(n) for n = 1..1000
G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A331350 (oriented), A063843 (unoriented), A331352 (chiral).
Other polychora: A331361 (8-cell), A331357 (16-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A327086 (simplex edges and ridges) and A337886 (simplex faces and peaks).

Mathematica
```
Table[(5 n^3 + n^7)/6, {n, 1, 25}]
```

Vec(x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8 + O(x^25)) \\ Colin Barker, Jan 15 2020

a(n) = (5*n^3 + n^7) / 6.
a(n) = C(n,1) + 26*C(n,2) + 306*C(n,3) + 1400*C(n,4) + 2800*C(n,5) + 2520*C(n,6) + 840*C(n,7), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A063843(n) - A331350(n) = A331350(n) - 2*A331352(n) = A063843(n) - A331352(n).
From Colin Barker, Jan 15 2020: (Start)
G.f.: x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 8200, 9080559, 1503323520, 81461669375, 2146080958056, 34228350856910, 377534786525184, 3140004522270465, 20896479183085000, 116094911796177061, 555622588428635520, 2346039511676401359, 8903083257215729960

Offset: 1

Robert A. Russell, Jan 14 2020

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the square faces of a tesseract {4,3,3} with n available colors.
There are 192 elements in the automorphism group of the 4-dimensional orthoplex that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  4          6      8x_2^2x_4^5
  31         8      4x_3^4x_6^2 + 4x_6^4
  22         3      8x_1^2x_2^1x_4^5
  211        6      2x_1^2x_2^11 + 2x_1^6x_2^9 + 4x_2^2x_4^5
  1111       1      4x_1^12x_2^6 + 4x_2^12

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Cf. A331354 (oriented), A331355 (unoriented), A331356 (chiral).
Other polychora: A331353 (5-cell), A331361 (8-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A337414 (orthoplex edges, orthotope ridges) and A337890 (orthotope faces, orthoplex peaks).

Table[(8n^4 + 8n^6 + 18n^7 + 6n^8 + n^12 + 3n^13 + 3n^15 + n^18)/48, {n, 1, 25}]

a(n) = (8*n^4 + 8*n^6 + 18*n^7 + 6*n^8 + n^12 + 3*n^13 + 3*n^15 + n^18) / 48.
a(n) = C(n,1) + 8198*C(n,2) + 9055962*C(n,3) + 1467050480*C(n,4) + 74035775370*C(n,5) + 1679679306420*C(n,6) + 20864180531565*C(n,7) + 159341117375160*C(n,8) + 804216787965360*C(n,9) + 2808560520334800*C(n,10) + 6981656802951600*C(n,11) + 12540346820971200*C(n,12) + 16328843044113600*C(n,13) + 15272715797539200*C(n,14) + 10003790644848000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A331355(n) - A331354(n) = A331354(n) - 2*A331356(n) = A331355(n) - A331356(n).

A331361 Number of achiral colorings of the edges of a tesseract with n available colors.

Original entry on oeis.org

1, 93024, 294157089, 91983927296, 7960001890625, 304914963625056, 6652124939544609, 96100248309858304, 1013293206632601441, 8334166666733500000, 56066328722011832961, 319495406392484665344

Offset: 1

Robert A. Russell, Jan 14 2020

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.
There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  4          6      8x_4^8
  31         8      4x_1^2x_3^2x_6^4 + 4x_2^1x_6^5
  22         3      8x_4^8
  211        6      2x_1^8x_2^12 + 2x_2^16 + 4x_4^8
  1111       1      4x_1^8x_2^12 + 4x_2^16

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. A331358 (oriented), A331359 (unoriented), A331360 (chiral).

Cf. A331353 (simplex), A331357 (orthoplex), A338955 (24-cell), A338967 (120-cell, 600-cell).

Table[(2n^6 + 8n^8 + n^16 + n^20)/12, {n, 1, 25}]

a(n) = (2*n^6 + 8*n^8 + n^16 + n^20) / 12.
a(n) = C(n,1) + 93022*C(n,2) + 293878020*C(n,3) + 90807857080*C(n,4) + 7503022894800*C(n,5) + 258528829444320*C(n,6) + 4681671089961600*C(n,7) + 50981530073846400*C(n,8) + 363246007692204000*C(n,9) + 1789536284820648000*C(n,10) + 6323058513173001600*C(n,11) + 16406578807069651200*C(n,12) + 31689737477798400000*C(n,13) + 45786987328642560000*C(n,14) + 49291621471572480000*C(n,15) + 38970361271761920000*C(n,16) + 21972146261345280000*C(n,17) + 8363100653107200000*C(n,18) + 1926047423139840000*C(n,19) + 202741834014720000*C(n,20), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A331359(n) - A331358(n) = A331358(n) - 2*A331360(n) = A331359(n) - A331360(n).

A338951 Number of achiral colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 6504, 8416440, 1455789440, 80139247500, 2125945744776, 34026498820524, 376045864704000, 3131319814422255, 20854395850585000, 115919421344402676, 554976171149122944, 2343894146343268610, 8896568181794053320

Offset: 1

Robert A. Russell, Nov 17 2020

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 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    Odd Cycle Indices     Count    Odd Cycle Indices
   x_1^12x_2^6             72     x_2^2x_4^5
   x_1^6x_2^9              96     x_1^2x_2^2x_6^3
   x_1^2x_2^11             96     x_2^3x_3^2x_6^2
   x_2^12                  96     x_3^4x_6^2
   x_1^2x_2^1x_4^5         96     x_6^4

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Cf. A338948 (oriented), A338949 (unoriented), A338950 (chiral), A338955 (edges, faces), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338967 (120-cell, 600-cell).

Table[(8n^4+8n^6+22n^7+6n^8+n^12+n^13+n^15+n^18)/48,{n,15}]

a(n) = (8*n^4 + 8*n^6 + 22*n^7 + 6*n^8 + n^12 + n^13 + n^15 + n^18) / 48.
a(n) = 1*C(n,1) + 6502*C(n,2) + 8396931*C(n,3) + 1422162700*C(n,4) + 72944399665*C(n,5) + 1666778870130*C(n,6) + 20777144613015*C(n,7) + 158973991255800*C(n,8) + 803196369526320*C(n,9) + 2806639981714800*C(n,10) + 6979192091902800*C(n,11) + 12538220293368000*C(n,12) + 16327662245294400*C(n,13) + 15272334392515200*C(n,14) + 10003736158416000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A338949(n) - A338948(n) = A338948(n) - 2*A338950(n) = A338949(n) - A338950(n).

A338952 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 137548893254081168086800768, 11046328890861011039111168376671536861388643, 10897746068379654103881579020805286236644252743361724416

Offset: 1

Robert A. Russell, Nov 17 2020

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. There are 576 elements in the rotation group of the 24-cell. They divide into 20 conjugacy classes. The first formula is obtained by averaging the edge (or face) 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_1^96              6+6+36+36   x_4^24
      72   x_1^4x_2^46                32   x_2^3x_6^15
    1+18   x_2^48                 8+8+32   x_6^16
      32   x_1^6x_3^30             72+72   x_8^12
  8+8+32   x_3^32                  48+48   x_12^8

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 97.

Cf. A338953 (unoriented), A338954 (chiral), A338955 (achiral), A338956 (exactly n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338964 (120-cell, 600-cell).

Table[(96n^8+144n^12+48n^16+32n^18+84n^24+48n^32+32n^36+19n^48+72n^50+n^96)/576,{n,15}]

a(n) = (96*n^8 + 144*n^12 + 48*n^16 + 32*n^18 + 84*n^24 + 48*n^32 + 32*n^36 + 19*n^48 + 72*n^50 + n^96) / 576.
a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).
a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).

A338953 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 68774446639102959610154176, 5523164445430505961199114292414803649442426, 5448873034189827051954635848284422749083650351583379456, 10956401461402941741829572441752281718329313621842215239237500000

Offset: 1

Robert A. Russell, Nov 17 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual.

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 97.

Cf. A338952 (oriented), A338954 (chiral), A338955 (achiral), A338957 (exactly n colors), A338949 (vertices, facets), A063843 (5-cell), A331359 (8-cell edges, 16-cell faces), A331355 (16-cell edges, 8-cell faces), A338965 (120-cell, 600-cell).

Table[(96n^8+144n^12+144n^16+128n^18+192n^20+228n^24+48n^32+32n^36+43n^48+72n^50+12n^52+12n^60+n^96)/1152,{n,15}]

a(n) = (96*n^8 + 144*n^12 + 144*n^16 + 128*n^18 + 192*n^20 + 228*n^24 +
48*n^32 + 32*n^36 + 43*n^48 + 72*n^50 + 12*n^52 + 12*n^60 + n^96) / 1152.
a(n) = Sum_{j=1..Min(n,96)} A338957(n) * binomial(n,j).
a(n) = A338952(n) - A338954(n) = (A338952(n) + A338955(n)) / 2 = A338954(n) + A338955(n).

A338954 Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

68774446614978208476646592, 5523164445430505077912054084256733211946217, 5448873034189827051926943172520863487560602391778344960, 10956401461402941741829554371669666304159415287557559324930859375

Offset: 2

Robert A. Russell, Nov 17 2020

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual.

Robert A. Russell, Table of n, a(n) for n = 2..30
Index entries for linear recurrences with constant coefficients, order 97.

Cf. A338952 (oriented), A338953 (unoriented), A338955 (achiral), A338958 (exactly n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338966 (120-cell, 600-cell).

Table[(96n^8+144n^12-48n^16-64n^18-192n^20-60n^24+48n^32+32n^36-5n^48+72n^50-12n^52-12n^60+n^96)/1152,{n,2,15}]

a(n) = (96*n^8 + 144*n^12 - 48*n^16 - 64*n^18 - 192*n^20 - 60*n^24 +
48*n^32 + 32*n^36 - 5*n^48 + 72*n^50 - 12*n^52 - 12*n^60 + n^96) / 1152.
a(n) = Sum_{j=2..Min(n,96)} A338958(n) * binomial(n,j).
a(n) = A338952(n) - A338953(n) = (A338952(n) - A338955(n)) / 2 = A338953(n) - A338955(n).

A338959 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

1, 24124751133507582, 883287060135783817036973460, 27692672230411020835164184856095160, 18069944152044184972628509749308321354400, 1018093811663859334508633754250963606821400320

Offset: 1

Robert A. Russell, Nov 17 2020

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>60, a(n) = 0.

Robert A. Russell, Table of n, a(n) for n = 1..60

Cf. A338956 (oriented), A338957 (unoriented), A338958 (chiral), A338955 (up to n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338983 (120-cell, 600-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (*binomial series*)
Drop[CoefficientList[bp[16]/6+bp[18]/6+bp[20]/3+bp[24]/4+bp[48]/24+bp[52]/48+bp[60]/48,x],1]

A338955(n) = Sum_{j=1..Min(n,60)} a(n) * binomial(n,j).

a(n) = 2*A338957(n) - A338956(n) = A338956(n) - 2*A338958(n) = A338957(n) - A338958(n).

cp's OEIS Frontend

A331353 Number of achiral colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

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A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331361 Number of achiral colorings of the edges of a tesseract with n available colors.

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A338951 Number of achiral colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

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A338952 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

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A338953 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

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A338954 Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

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A338959 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

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