A128766 -id:A128766 - cp's OEIS frontend

A199406 The number of inequivalent ways to color the edges of a cube using at most n colors.

1, 144, 12111, 358120, 5131650, 45528756, 288936634, 1433251296, 5887880415, 20842168600, 65402344161, 185788177224, 485443851256, 1181242399260, 2703252560100, 5864398969216, 12138503871789, 24101498435616, 46112016365155, 85335258695400, 153249227870046

Offset: 1

Geoffrey Critzer, Nov 05 2011

Two edge colorings are equivalent if one is the mirror image of the other or the cube can be picked up and rotated in any manner to obtain the other.
The group here has order 48 (compare A060530). - N. J. A. Sloane, Aug 14 2012
Also the number of unoriented colorings of the 12 edges of a regular octahedron with n or fewer colors. The Schläfli symbols of the cube and octahedron are {4,3} and {3,4} respectively. They are mutually dual. For an unoriented coloring, chiral pairs are counted as one. - Robert A. Russell, Oct 17 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A060530 (oriented), A337406 (chiral), A331351 (achiral), A128766 (cube vertices, octahedron faces), A198833 (cube faces, octahedron vertices), A063842(n-1) (tetrahedron), A337963 (dodecahedron, icosahedron).

Row 3 of A337408 (orthotope edges, orthoplex ridges) and A337412 (orthoplex edges, orthotope ridges).

Table[CycleIndex[KSubsetGroup[Automorphisms[CubicalGraph], Edges[CubicalGraph]],s] /. Table[s[i]->n, {i,1,6}], {n,1,15}]
Table[(8n^2+12n^3+8n^4+4n^6+12n^7+3n^8+n^12)/48, {n,20}] (* Robert A. Russell, Oct 17 2020 *)

a(n) = n^12/48 + n^8/16 + n^7/4 + n^6/12 + n^4/6 + n^3/4 + n^2/6.
Cycle index = 1/48(s_1^12+3s_1^4s_2^4+12s_1^2s_2^5+4s_2^6+8s_3^4+12s_4^3+8s_6^2).
G.f.: -x*(76*x^10 +10016*x^9 +212772*x^8 +1380453*x^7 +3384939*x^6 +3388593*x^5 +1380279*x^4 +211623*x^3 +10317*x^2 +131*x +1)/(x -1)^13. [Colin Barker, Aug 13 2012]
From Robert A. Russell, Oct 17 2020: (Start)
a(n) = A060530(n) - A337406(n) = (A060530(n) + A331351(n)) / 2 = A337406(n) + A331351(n).
a(n) = 1*C(n,1) + 142*C(n,2) + 11682*C(n,3) + 310536*C(n,4) + 3460725*C(n,5) + 19870590*C(n,6) + 65886660*C(n,7) + 133585200*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors. (End)

A000543 Number of inequivalent ways to color vertices of a cube using at most n colors.

Original entry on oeis.org

0, 1, 23, 333, 2916, 16725, 70911, 241913, 701968, 1798281, 4173775, 8942021, 17930628, 34009053, 61518471, 106823025, 179003456, 290715793, 459239463, 707740861, 1066780100, 1576090341, 2286660783, 3263156073, 4586706576

Offset: 0

Clint. C. Williams (Clintwill(AT)aol.com)

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the vertices has cycle index (x1^8 + 9*x2^4 + 6*x4^2 + 8*x1^2*x3^2)/24.
Also the number of ways to color the faces of a regular octahedron with n colors, counting mirror images separately.
From Robert A. Russell, Oct 08 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
There are 24 elements in the rotation group of the regular octahedron/cube. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class    Count    Even Cycle Indices
  Identity              1     x_1^8
  Vertex rotation       8     x_1^2x_3^2
  Edge rotation         6     x_2^4
  Small face rotation   6     x_4^2
  Large face rotation   3     x_2^4 (End)

N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see p. 147).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Polyhedron Coloring
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A128766 (unoriented), A337896 (chiral), A337897 (achiral).
Other elements: A060530 (edges), A047780 (cube faces, octahedron vertices).
Cf. A006008 (tetrahedron), A000545 (dodecahedron faces, icosahedron vertices), A054472 (icosahedron faces, dodecahedron vertices).
Row 3 of A325012 (orthotope vertices, orthoplex facets) and A337891 (orthoplex faces, orthotope peaks).

[(1/24)*n^2*(n^6+17*n^2+6): n in [0..30]]; // Vincenzo Librandi, Apr 15 2012

f:= n->(1/24)*n^2*(n^6+17*n^2+6); seq(f(n), n=0..40);

CoefficientList[Series[x*(1+x)*(1+13*x+149*x^2+514*x^3+149*x^4+13*x^5+x^6)/(1-x)^9,{x,0,30}],x] (* Vincenzo Librandi, Apr 15 2012 *)
Table[(n^8+17n^4+6n^2)/24,{n,0,30}] (* Robert A. Russell, Oct 08 2020 *)

a(n) = (1/24)*n^2*(n^6+17*n^2+6). (Replace all x_i's in the cycle index with n.)
G.f.: x*(1+x)*(1+13*x+149*x^2+514*x^3+149*x^4+13*x^5+x^6)/(1-x)^9. - Colin Barker, Jan 29 2012
a(n) = 1*C(n,1) + 21*C(n,2) + 267*C(n,3) + 1718*C(n,4) + 5250*C(n,5) + 7980*C(n,6) + 5880*C(n,7) + 1680*C(n,8), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A128766(n) + A337896(n) = 2*A128766(n) - A337897(n) = 2*A337896(n) + A337897(n). - Robert A. Russell, Oct 08 2020

Entry revised by N. J. A. Sloane, Jan 03 2005

A252705 The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.

Original entry on oeis.org

1, 82, 5379, 148648, 2085655, 18356514, 116081245, 574795936, 2359033605, 8345970370, 26180606287, 74354990568, 194253329803, 472634761522, 1081541381145, 2346163937920, 4856060529001, 9641643580530, 18446420258299, 34136541925480, 61303301959263

Offset: 1

Robert A. Russell, Dec 20 2014

The cycle index using the full automorphism group for faces of a dodecahedron is (x1^12+15*x2^6+20*x3^4+24*x1^2*x5^2+15*x1^4*x2^4+x2^6+20*x6^2+24*x2*x10)/120.

Also the number of ways to color the vertices of a regular icosahedron with n colors, counting mirror images as one.

			For n=2, a(2)=82, the number of ways to color the faces of a regular dodecahedron with two colors, counting mirror images as the same. Of these, two use the same color for all faces, and 80 use both colors.

F. S. Roberts and B. Tesman, Applied Combinatorics, 2d Ed., Pearson Prentice Hall, 2005, pages 439-488.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992, pages 461-474.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000545 (number when mirror images are counted separately).

Cf. A000332 (tetrahedron), A198833 (cube), A128766 (octahedron), A252704 (icosahedron).

Table[n^2(n^2+1)(n^8-n^6+16n^4+44)/120,{n,1,30}]

vector(60, n, n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120) \\ Michel Marcus, Dec 21 2014

a(n) = n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120.
G.f.: x*(x+1)*(x^10+68*x^9+4323*x^8+80508*x^7+469548*x^6+886944*x^5+469548*x^4 +80508*x^3+4323*x^2+68*x+1)/(1-x)^13.
a(n) = C(n,1)+80*C(n,2)+5136*C(n,3)+127620*C(n,4)+1395390*C(n,5)+7965948*C(n,6) +26368272*C(n,7)+53438112*C(n,8)+67359600*C(n,9)+51559200*C(n,10)+21954240*C(n,11)+3991680*C(n,12). Each term indicates the number of ways to use n colors to color the dodecahedron with exactly 1, 2, 3, ..., 10, 11, or 12 colors.

A252704 The number of ways to color the faces of a regular icosahedron with n colors, counting mirror images as one.

Original entry on oeis.org

1, 9436, 29131965, 9164844880, 794760482005, 30468267440892, 664937321266057, 9607687940954944, 101313914601247929, 833333459683337020, 5606250353568935653, 31948001059902168528, 158374701054784400173, 697235469002925659548

Offset: 1

Robert A. Russell, Dec 20 2014

The cycle index using the full automorphism group for faces of an icosahedron is (x1^20+15*x2^10+20*x1^2*x3^6+24*x5^4+15*x1^4*x2^8+x2^10+20*x2*x6^3+24*x10^2)/120.

Also the number of ways to color the vertices of a regular dodecahedron with n colors, counting mirror images as one.

			For n=2, a(2)=9436, the number of ways to color the faces of a regular icosahedron with two colors, counting mirror images as the same. Of these, two use the same color for all faces, and 9434 use both colors.

F. S. Roberts and B. Tesman, Applied Combinatorics, 2d Ed., Pearson Prentice Hall, 2005, pages 439-488.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992, pages 461-474.

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. A054472 (number when mirror images are counted separately).

Cf. A000332 (tetrahedron), A198833 (cube), A128766 (octahedron), A252705 (dodecahedron).

Table[n^2(n^18+15n^10+16n^8+20n^6+44n^2+24)/120,{n,1,30}]

a(n) = n^2*(n^18+15*n^10+16*n^8+20*n^6+44*n^2+24)/120.
G.f.: x*(x+1)*(x^18+9414*x^17+28924605*x^16+8526129240*x^15+599877779040*x^14 +15064347905208*x^13+164923977484392*x^12+874644240573864*x^11 +2363591146376826*x^10+3299427410370820*x^9+2363591146376826*x^8 +874644240573864*x^7+164923977484392*x^6+15064347905208*x^5 +599877779040*x^4+8526129240*x^3+28924605*x^2+9414*x+1)/(1-x)^21.
a(n) = C(n,1)+9434*C(n,2)+29103660*C(n,3)+9048373632*C(n,4)+749227482900*C(n,5) +25836594724296*C(n,6)+468029669151744*C(n,7)+5097434180194944*C(n,8) +36322119730219680*C(n,9)+178947770105039040*C(n,10)+632296226073536640*C(n,11)+1640646875234062080*C(n,12)+3168965153453299200*C(n,13)+4578694359419980800*C(n,14)+4929160839482880000*C(n,15)+3897035952819609600*C(n,16) +2197214626134528000*C(n,17)+836310065310720000*C(n,18)+192604742313984000*C(n,19)+20274183401472000*C(n,20).  Each term indicates the number of ways to use n colors to color the icosahedron with exactly 1, 2, 3, ..., 18, 19, or 20 colors.

A325013 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 10, 21, 22, 1, 15, 55, 267, 402, 1, 21, 120, 1996, 132102, 1228158, 1, 28, 231, 10375, 11756666, 484086357207, 400507806843728, 1, 36, 406, 41406, 405385550, 4805323147589984, 74515759884862073604656433, 527471432057653004017274030725792, 1

Offset: 1

Robert A. Russell, May 27 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.

Also the number of unoriented colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

			Array begins with A(1,1):
1   3      6       10        15         21          28           36 ...
1   6     21       55       120        231         406          666 ...
1  22    267     1996     10375      41406      135877       384112 ...
1 402 132102 11756666 405385550 7416923886 86986719477 735192450952 ...
For A(2,2)=6, two squares have all edges the same color, two have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

Robert A. Russell, Table of n, a(n) for n = 1..78
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
Wikipedia, Cross-polytope

Cf. A325012 (oriented), A325014 (chiral), A325015 (achiral), A325017 (exactly k colors).
Other n-dimensional polytopes: A325000 (simplex), A325005 (orthotope).
Rows 1-4 are A000217, A002817, A128766, A128767; column 2 is A000616.
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[((CI0[#] + CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = A325012(n,k) - A325014(n,k) = (A325012(n,k) + A325015(n,k)) / 2 = A325014(n,k) + A325015(n,k).
A(n,k) = Sum_{j=1..2^n} A325017(n,j) * binomial(k,j).

A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.

Original entry on oeis.org

1, 10, 56, 220, 680, 1771, 4060, 8436, 16215, 29260, 50116, 82160, 129766, 198485, 295240, 428536, 608685, 848046, 1161280, 1565620, 2081156, 2731135, 3542276, 4545100, 5774275, 7268976, 9073260, 11236456, 13813570, 16865705, 20460496, 24672560, 29583961

Offset: 1

Geoffrey Critzer, Oct 30 2011

The cycle index: 1/48 (s_1^6 + 3 s_1^4 s_2 + 9 s_1^2 s_2^2 +7 s_2^3 + 8 s_3^2 + 6 s_1^2 s_4 + 6 s_2 s_4 + 8 s_6) is returned in Mathematica by CycleIndex[ Automorphisms[ OctahedralGraph ], s].
One-sixth the area of the right triangles with sides 2b+2, b^2+2b, and b^2+2b+2 with b = A000217(n), the n-th triangular number. - J. M. Bergot, Aug 02 2013
Also the number of ways to color the faces of a cube with n colors, counting each pair of mirror images as one.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Cf. A047780 (oriented), A093566(n+1) (chiral), A337898 (achiral), A199406 (edges), A128766 (octahedron faces, cube vertices), A000332(n+3) (tetrahedron), A128766 (octahedron faces, cube vertices), A252705 (dodecahedron faces, icosahedron vertices), A252704 (icosahedron faces, dodecahedron vertices), A000217 (triangular numbers).

Row 3 of A325005 (orthotope facets, orthoplex vertices) and A337888 (orthotope faces, orthoplex peaks).

[n*(n+1)*(n^2+n+2)*(n^2+n+4)/48: n in [1..35]]; // Vincenzo Librandi, Aug 04 2013

Table[(n^6 + 3 n^5 + 9 n^4 + 13 n^3 + 14 n^2 + 8 n)/48, {n, 25}]
CoefficientList[Series[-(1 + 3 x + 7 x^2 + 3 x^3 + x^4) / (x - 1)^7, {x, 0, 35}], x] (* Vincenzo Librandi, Aug 04 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,56,220,680,1771,4060},40] (* Harvey P. Dale, Nov 06 2024 *)

a(n)=n*(n+1)*(n^2+n+2)*(n^2+n+4)/48 \\ Charles R Greathouse IV, Aug 02 2013

a(n) = n*(n+1)*(n^2+n+2)*(n^2+n+4)/48.
G.f.: x*(1+3*x+7*x^2+3*x^3+x^4) / (1-x)^7. - R. J. Mathar, Oct 30 2011
a(n) = Sum_{i=1..A000217(n)} A000217(i). [Bruno Berselli, Sep 06 2013]
a(n) = 1*C(n,1) + 8*C(n,2) + 29*C(n,3) + 52*C(n,4) + 45*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A047780(n) - A093566(n+1) = (A047780(n) + A337898(n)) / 2 = A093566(n+1) + A337898(n). - Robert A. Russell, Oct 19 2020

A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

Original entry on oeis.org

1, 21, 201, 1076, 4025, 11901, 29841, 66256, 134001, 251725, 445401, 750036, 1211561, 1888901, 2856225, 4205376, 6048481, 8520741, 11783401, 16026900, 21474201, 28384301, 37055921, 47831376, 61100625, 77305501, 96944121

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^4
  Vertex rotation*       8     x_2^1x_6^1      Asterisk indicates that the
  Edge rotation*         6     x_1^4x_2^2      operation is followed by an
  Small face rotation*   3     x_4^2           inversion.
  Large face rotation*   6     x_2^4

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).
Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).
Other polyhedra: A006003 (tetrahedron), A337962  (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).

Mathematica
```
Table[n^2(7+2n^2+3n^4)/12, {n,30}]
```

a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.
a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).
G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.

A337896 Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

Original entry on oeis.org

0, 1, 66, 920, 6350, 29505, 106036, 317856, 832140, 1961025, 4248310, 8590296, 16398746, 29814785, 51983400, 87399040, 142333656, 225359361, 347978730, 525376600, 777308070, 1129138241, 1613050076, 2269437600

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

Each member of a chiral pair is a reflection, but not a rotation, of the other.

			For a(2)=1, centering the octahedron (cube) at the origin and aligning the diagonals (edges) with the axes, color the faces (vertices) in the octants ---, --+, -++, and +++ with one color and the other 4 elements with the other color.

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A000543 (oriented), A128766(unoriented), A337897 (achiral).
Other elements: A337406 (edges), A093566(n+1) (cube faces, octahedron vertices).
Other polyhedra: A000332 (simplex), A093566(n+1) (cube/octahedron).
Row 3 of A325014 (chiral pairs of colorings of orthoplex facets or orthotope vertices).
Row 3 of A337893 (chiral pairs of colorings of orthoplex faces or orthotope peaks).

Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n,30}]

a(n) = (n-1) * n^2 * (n+1) * (8 - 5*n^2 + n^4) / 48.
a(n) = 1*C(n,2) + 63*C(n,3) + 662*C(n,4) + 2400*C(n,5) + 3900*C(n,6) + 2940*C(n,7) + 840*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
G.f.: x^2 * (1+x) * (1+56*x+306*x^2+56*x^3+x^4) / (1-x)^9.
a(n) = A000543(n) - A128766(n) = (A000543(n) - A337897(n)) / 2 = A128766(n) - A337897(n).

A337892 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 22, 1, 4, 267, 11251322, 1, 5, 1996, 4825746875682, 314824532572147370464, 1, 6, 10375, 48038446526132256, 38491882660671134164965704408524083, 31716615393638864931753532641338560302264320, 1

Offset: 2

Robert A. Russell, Sep 28 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. For n=2, the figure is a square with one square face. For n=3, the figure is an octahedron with 8 triangular faces. For higher n, the number of triangular faces is 8*C(n,3).

Also the number of unoriented colorings of the peaks of an n-dimensional orthotope (hypercube). A peak is an (n-3)-dimensional orthotope.

			Array begins with T(2,1):
 1        2             3                 4                    5 ...
 1       22           267              1996                10375 ...
 1 11251322 4825746875682 48038446526132256 60632984344185045000 ...

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

Cf. A337891 (oriented), A337893 (chiral), A337894 (achiral).
Other elements: A325005 (vertices), A337412 (edges).
Other polytopes: A337884 (simplex), A337888 (orthotope).
Rows 2-4 are A000027, A128766, A331359

m=2; (* dimension of color element, here a face *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, m+1]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[]);
PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
row[m]=b;
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^n)]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,6}, {n,m,d+m-1}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = A337891(n,k) - A337893(n,k) = (A337891(n,k) + A337894(n,k)) / 2 = A337893(n,k) + A337894(n,k).

A378473 The number of n-colorings of the vertices of the truncated octahedron up to rotation and reflection.

Original entry on oeis.org

0, 1, 355048, 5886817533, 5864336054656, 1241773051013125, 98716454926955496, 3991277735434713913, 98382652674879674368, 1661801013342756245961, 20833333958666683585000, 205202766952229526577141, 1656184328295547539616128, 11308349424395689922231053

Offset: 0

Peter Kagey, Nov 27 2024

nonn

Equivalently, the number of n-colorings of the faces of the tetrakis hexahedron, which is the polyhedral dual of the truncated octahedron.

Colorings are counted up to the full octahedral group of order 48.

Wikipedia, Tetrakis hexahedron
Wikipedia, Truncated octahedron

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378474, A378475, A378476, A378477, A378478.

a(n) = (1/48)*(n^24 + 3*n^16 + 16*n^12 + 8*n^8 + 12*n^6 + 8*n^4).

Asymptotically, a(n) ~ n^24/48.

cp's OEIS Frontend

A199406 The number of inequivalent ways to color the edges of a cube using at most n colors.

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A000543 Number of inequivalent ways to color vertices of a cube using at most n colors.

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A252705 The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.

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A252704 The number of ways to color the faces of a regular icosahedron with n colors, counting mirror images as one.

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A325013 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

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A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.

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A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

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