A337896 -id:A337896 - cp's OEIS frontend

A093566 a(n) = n(n-1)(n-2)(n-3)(n^2-3*n-2)/48.

0, 0, 0, 0, 1, 20, 120, 455, 1330, 3276, 7140, 14190, 26235, 45760, 76076, 121485, 187460, 280840, 410040, 585276, 818805, 1125180, 1521520, 2027795, 2667126, 3466100, 4455100, 5668650, 7145775, 8930376, 11071620, 13624345, 16649480, 20214480

Offset: 0

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

a(n+1) is the number of chiral pairs of colorings of the faces of a cube (vertices of a regular octahedron) using n or fewer colors. - Robert A. Russell, Sep 28 2020

			For a(3+1) = 1, each of the three colors is applied to a pair of adjacent faces of the cube (vertices of the octahedron). - _Robert A. Russell_, Sep 28 2020

Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

From Robert A. Russell, Sep 28 2020: (Start)
Cf. A047780 (oriented), A198833 (unoriented), A337898 (achiral) colorings.
a(n+1) = A325006(3,n) (chiral pairs of colorings of orthotope facets or orthoplex vertices).
a(n+1) = A337889(3,n) (chiral pairs of colorings of orthotope faces or orthoplex peaks).
Other polyhedra: A000332 (tetrahedron), A337896 (cube/octahedron).
(End)

Table[ Binomial[ Binomial[n-1, 2], 3], {n,0,32}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,1,20,120},40] (* Harvey P. Dale, Feb 18 2016 *)

a(n)=n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48 \\ Charles R Greathouse IV, Jun 11 2015

[(binomial(binomial(n,2),3)) for n in range(-1, 33)] # Zerinvary Lajos, Nov 30 2009

a(n) = binomial(binomial(n-1, 2), 3).
G.f.: -x^4*(1+13*x+x^2)/(x-1)^7. - R. J. Mathar, Dec 08 2010
a(n+1) = 1*C(n,3) + 16*C(n,4) + 30*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors. - Robert A. Russell, Sep 28 2020
a(n) = A000217(n-1)*A239352(n-2)/6. - R. J. Mathar, Mar 25 2022

Edited (with a new definition) by N. J. A. Sloane, Jul 02 2008

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

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.

A337893 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 66, 11158298, 0, 0, 920, 4825452718593, 314824333015938998688, 0, 0, 6350, 48038354542204960, 38491882659300767730994725249684096, 31716615393292685397985382790580028572676096, 0

Offset: 2

Robert A. Russell, Sep 28 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. 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 chiral pairs of colorings of the peaks of an n-dimensional orthotope (hypercube). A peak is an (n-3)-dimensional orthotope.

			Table begins with T(2,1):
 0        0             0                 0                    0 ...
 0        1            66               920                 6350 ...
 0 11158298 4825452718593 48038354542204960 60632976384183154375 ...

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), A337892 (unoriented), A337894 (achiral).
Other elements: A325006 (vertices), A337413 (edges).
Other polytopes: A337885 (simplex), A337889 (orthotope).
Rows 2-4 are A000004, A337896, A331360.

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}]; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],1,-1]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[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) - A337892(n,k) = (A337891(n,k) - A337894(n,k)) / 2 = A337892(n,k) - A337894(n,k).

A337961 Number of chiral pairs of colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.

Original entry on oeis.org

0, 14, 3720, 132184, 1987720, 17935806, 114638048, 570597216, 2348263008, 8320953630, 26126986952, 74247445272, 194049316552, 472265688622, 1080900468480, 2345089916288, 4854316187136, 9638888023278, 18442173583176

Offset: 1

Robert A. Russell, Oct 03 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

Harvey P. Dale, 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 (oriented), A252705 (unoriented), A337962 (achiral).
Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337964 (edges).
Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Table[(n^12-15n^8+14n^6+44n^4-44n^2)/120,{n,30}]
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,14,3720,132184,1987720,17935806,114638048,570597216,2348263008,8320953630,26126986952,74247445272,194049316552},20] (* Harvey P. Dale, Nov 17 2024 *)

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 14*n^4 + 44) / 120.
a(n) = 14*C(n,2) + 3678*C(n,3) + 117388*C(n,4) + 1363860*C(n,5) + 7918056*C(n,6) + 26332992*C(n,7) + 53428032*C(n,8) + 67359600*C(n,9) + 51559200*C(n,10) + 21954240*C(n,11) + 3991680*C(n,12), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A000545(n) - A252705(n) = (A000545(n) - A337962(n)) / 2 = A252705(n) - A337962(n).

A337959 Number of chiral pairs of colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

Original entry on oeis.org

0, 8388, 28998090, 9160633008, 794699283870, 30467722237092, 664933856235516, 9607670743188672, 101313843935748516, 833333209516666980, 5606249568529546134, 31947998829845093424, 158374695227965468434

Offset: 1

Robert A. Russell, Oct 03 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

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 (oriented), A252704 (unoriented), A337960 (achiral).
Other elements: A337964 (edges), A337961 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Table[(n^20-15n^12+14n^10+20n^8+4n^4-24n^2)/120,{n,30}]

a(n) = (n-1) * n^2 * (n+1) * (n^2+2) * (n^14 - n^12 + 3*n^10 - 5*n^8 - 4*n^6 + 8*n^4 + 4*n^2 + 12) /120.
a(n) = 8388*C(n,2) + 28972926*C(n,3) + 9044690976*C(n,4) + 749186015850*C(n,5) + 25836356193012*C(n,6) + 468028878138864*C(n,7) + 5097432576698784*C(n,8) + 36322117709159520*C(n,9) + 178947768558202560*C(n,10) + 632296225414909440*C(n,11) + 1640646875114311680*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), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A054472(n) - A252704(n) = (A054472(n) - A337960(n)) / 2 = A252704(n) - A337960(n).

cp's OEIS Frontend

A093566 a(n) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48.

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A000543 Number of inequivalent ways to color vertices of a cube 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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A337893 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A337961 Number of chiral pairs of colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.

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A337959 Number of chiral pairs of colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

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A093566 a(n) = n(n-1)(n-2)(n-3)(n^2-3*n-2)/48.