A337406 -id:A337406 - cp's OEIS frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

0, 1, 218, 22815, 703760, 10194250, 90775566, 576941778, 2863870080, 11769161895, 41669295250, 130772947481, 371513523888, 970769847320, 2362273657030, 5406141568500, 11728193258496, 24276032182173, 48201464902410, 92221684354915

Offset: 0

N. J. A. Sloane, Apr 11 2001

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the edges has cycle index (x1^12 + 3*x2^6 + 6*x4^3 + 6*x1^2*x2^5 + 8*x3^4)/24.
Also, number of inequivalent colorings of the edges of a regular octahedron using at most n colors. - José H. Nieto S., Jan 19 2012
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 edge 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_1^12
  Vertex rotation       8     x_3^4
  Edge rotation         6     x_1^2x_2^5
  Small face rotation   6     x_4^3
  Large face rotation   3     x_2^6 (End)
Also, number of ways of coloring the vertices of the truncated tetrahedron or faces of the triakis tetrahedron up to rotation and reflection. - Peter Kagey, Nov 27 2024

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).

Harry J. Smith, Table of n, a(n) for n=0..200
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A199406 (unoriented), A337406 (chiral), A331351 (achiral).
Other elements: A000543 (cube vertices, octahedron faces), A047780 (cube faces, octahedron vertices).
Cf. A046023 (tetrahedron), A282670 (dodecahedron/icosahedron).
Row 3 of A337407 (orthotope edges, orthoplex ridges) and A337411 (orthoplex edges, orthotope ridges).

Table[(n^12+6n^7+3n^6+8n^4+6n^3)/24,{n,0,20}] (* Harvey P. Dale, Feb 13 2013 *)

{ for (n=0, 200, write("b060530.txt", n, " ", (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24); ) } \\ Harry J. Smith, Jul 06 2009

a(n) = (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24. (Replace all x_i's in the cycle index by n.)
G.f.: -x*(150*x^10 +19758*x^9 +425032*x^8 +2763481*x^7 +6769435*x^6 +6773089*x^5 +2763307*x^4 +423883*x^3 +20059*x^2 +205*x +1)/(x -1)^13. - Colin Barker, Aug 13 2012
From Robert A. Russell, Oct 08 2020: (Start)
a(n) = 1*C(n,1) + 216*C(n,2) + 22164*C(n,3) + 613804*C(n,4) + 6901425*C(n,5) + 39713430*C(n,6) + 131754420*C(n,7) + 267165360*C(n,8) + 336798000*C(n,9) + 257796000*C(n,10) + 109771200*C(n,11) + 19958400*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A199406(n) + A337406(n) = 2*A199406(n) - A331351(n) = 2*A337406(n) + A331351(n). (End)

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

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

Original entry on oeis.org

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)

A337409 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 74, 0, 0, 15, 10704, 11158298, 0, 0, 45, 345640, 4825452718593, 314824408633217132928, 0, 0, 105, 5062600, 48038354542204960, 38491882659952177472606694634030116, 136221825854745676076981182469325427379054390050209792, 0

Offset: 1

Robert A. Russell, Aug 26 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is a cube with 12 edges. The number of edges is n*2^(n-1).

Also the number of chiral pairs of colorings of the regular (n-2)-dimensional simplexes in a regular n-dimensional orthoplex.

			Table begins with T(1,1):
0  0     0      0       0        0         0          0          0 ...
0  0     3     15      45      105       210        378        630 ...
0 74 10704 345640 5062600 45246810 288005144 1430618784 5881281480 ...
For T(2,3)=3, the chiral arrangements are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.

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. A337407 (oriented), A337408 (unoriented), A337410 (achiral).
Rows 2-4 are A050534, A337406, A331360.
Cf. A327085 (simplex edges), A337413 (orthoplex edges), A325014 (orthotope vertices).

m=1; (* dimension of color element, here an edge *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1+2x^(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, n-m]];
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,7}, {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) = A337407(n,k) - A337408(n,k) = (A337407(n,k) - A337410(n,k)) / 2 = A337408(n,k) - A337410(n,k).

A337413 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 74, 0, 0, 15, 10704, 40927, 0, 0, 45, 345640, 731279799, 280317324, 0, 0, 105, 5062600, 732272925320, 3163614120031068, 24869435516248, 0, 0, 210, 45246810, 155180061396500, 314800331906964016128, 919853357924272852197243, 29931599129719666392, 0

Offset: 1

Robert A. Russell, Aug 26 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is an octahedron with 12 edges. The number of edges is 2n*(n-1) for n>1.

Also the number of chiral pairs of colorings of the regular (n-2)-dimensional orthotopes (hypercubes) in a regular n-dimensional orthotope.

			Table begins with T(1,1):
0  0     0      0       0        0         0          0          0 ...
0  0     3     15      45      105       210        378        630 ...
0 74 10704 345640 5062600 45246810 288005144 1430618784 5881281480 ...
For T(2,3)=3, the chiral arrangements are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.

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. A337411 (oriented), A337412 (unoriented), A337414 (achiral).
Rows 2-4 are A050534, A337406, A331356.
Cf. A327085 (simplex edges), A337409 (orthotope edges), A325006 (orthoplex vertices).

m=1; (* dimension of color element, here an edge *)
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,8}, {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) = A337411(n,k) - A337412(n,k) = (A337411(n,k) - A337414(n,k)) / 2 = A337412(n,k) - A337414(n,k).

A331351 Number of achiral colorings of the edges of a cube or regular octahedron.

Original entry on oeis.org

1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935, 15041950, 31740841, 62830560, 117855192, 211141490, 363551700, 604679936, 975561405, 1531968822, 2348375395, 3522668800, 5181705606, 7487800650, 10646250902

Offset: 1

Robert A. Russell, Jan 14 2020

A cube has 8 vertices and 12 edges. A regular octahedron has 6 vertices and 12 edges. An achiral coloring is identical to its reflection.
From Robert A. Russell, Oct 08 2020: (Start)
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 edge 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_2^6
  Vertex rotation*       8     x_6^2            Asterisk indicates that the
  Edge rotation*         6     x_1^2x_2^5       operation is followed by an
  Small face rotation*   3     x_4^3            inversion.
  Large face rotation*   6     x_1^4x_2^4 (End)

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A060530 (oriented), A199406 (unoriented), A337406 (chiral), A337897 (octahedron faces, cube vertices), A337898 (cube faces, octahedron vertices), A037270 (tetrahedron), A337953 (dodecahedron, icosahedron).

Row 3 of A337410 (orthotope edges, orthoplex ridges) and A337414 (orthoplex edges, orthotope ridges).

Table[(8n^2 + 6n^3 + n^6 + 6n^7 + 3n^8)/24, {n, 1, 30}]
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935}, 25]

a(n) = (8*n^2 + 6*n^3 + n^6 + 6*n^7 + 3*n^8) / 24.
a(n) = 1*C(n,1) + 68*C(n,2) + 1200*C(n,3) + 7268*C(n,4) + 20025*C(n,5) + 27750*C(n,6) + 18900*C(n,7) + 5040*C(n,8), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A199406(n) - A060530(n) = A060530(n) - 2*A337406(n) = A199406(n) - A337406(n). - Robert A. Russell, Oct 08 2020
G.f.: (x + 61*x^2 + 813*x^3 + 2253*x^4 + 1628*x^5 + 282*x^6 + 2*x^7) / (1-x)^9.
E.g.f.: (1/24)*exp(x)*x*(24 + 816*x + 4800*x^2 + 7268*x^3 + 4005*x^4 + 925*x^5 + 90*x^6 + 3*x^7). - Stefano Spezia, Jan 17 2020

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).

A337964 Number of chiral pairs of colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

Original entry on oeis.org

0, 8939560, 1715748562809, 9607677585671872, 7761021378582359350, 1842282662572342834488, 187827835730804603558945, 10316166993798251995440640, 353259652291613627252061348

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 (31, -465, 4495, -31465, 169911, -736281, 2629575, -7888725, 20160075, -44352165, 84672315, -141120525, 206253075, -265182525, 300540195, -300540195, 265182525, -206253075, 141120525, -84672315, 44352165, -20160075, 7888725, -2629575, 736281, -169911, 31465, -4495, 465, -31, 1).

Cf. A282670 (oriented), A337963 (unoriented), A337953 (achiral).
Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337961 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A337899 (tetrahedron), A337406 (cube/octahedron).

Table[(n^30-15n^17+15n^16-n^15+20n^10+24n^6-20n^5-24n^3)/120,{n,30}]

a(n) = (n^30 - 15*n^17 + 15*n^16 - n^15 + 20*n^10 + 24*n^6 - 20*n^5 - 24*n^3) / 120.
a(n) = 8939560*C(n,2) + 1715721744129*C(n,3) + 9600814645057996*C(n,4) + 7713000148050232480*C(n,5) + 1795860615149796593688*C(n,6) + 175094502333083946715914*C(n,7) + 8864694277747989482032560*C(n,8) + 267022176368352696363194640*C(n,9) + 5242809910438322709320514240*C(n,10) + 71533267863137818750780447680*C(n,11) + 710438037081549637823404041600*C(n,12) + 5315930749209804729425000380800*C(n,13) + 30757743469720886648597337369600*C(n,14) + 140355611183197552206530379513600*C(n,15) + 512749946932635113438921952768000*C(n,16) + 1516429386147442831718766368256000*C(n,17) + 3659586727743885232600161343488000*C(n,18) + 7243809192262705479647976345600000*C(n,19) + 11790166608014659213935198412800000*C(n,20) + 15777861864770715186138442260480000*C(n,21) + 17309780658863308912305163714560000*C(n,22) + 15473267984805657314364466790400000*C(n,23) + 11155559298200256484274739609600000*C(n,24) + 6385716995478673633837056000000000*C(n,25) + 2834140845518322325537731379200000*C(n,26) + 939989821959452064042418176000000*C(n,27) + 219202016094796777623060480000000*C(n,28) + 32051387227306419585220608000000*C(n,29) + 2210440498434925488635904000000*C(n,30), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A282670(n) - A337963(n) = (A282670(n) - A337953(n)) / 2 = A337963(n) - A337953(n).

A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

Original entry on oeis.org

0, 1, 21, 140, 575, 1785, 4606, 10416, 21330, 40425, 71995, 121836, 197561, 308945, 468300, 690880, 995316, 1404081, 1943985, 2646700, 3549315, 4694921, 6133226, 7921200, 10123750, 12814425, 16076151, 20001996

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. A regular tetrahedron has 6 edges and Schläfli symbol {3,3}.

			For a(2)=1, two opposite edges and one edge connecting those have one color; the other three edges have the other color.

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

Cf. A046023(unoriented), A063842(n-1) (oriented), A037270 (chiral).
Other elements: A000332 (vertices and faces).
Other polyhedra: A337406 (cube/octahedron).
Row 3 of A327085 (chiral pairs of colorings of edges or ridges of an n-simplex).

Mathematica
```
Table[(n-1)n^2(n+1)(n^2-2)/24, {n, 40}]
```

a(n) = (n-1) * n^2 * (n+1) * (n^2-2) / 24.
a(n) = 1*C(n,2) + 18*C(n,3) + 62*C(n,4) + 75*C(n,5) + 30*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A046023(n) - A063842(n-1) = (A046023(n) - A037270(n)) / 2 = A063842(n-1) - A037270(n).
G.f.: x^2 * (1+x) * (1+13x+x^2)/(1-x)^7.

cp's OEIS Frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

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

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A337409 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

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A337413 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A331351 Number of achiral colorings of the edges of a cube or regular octahedron.

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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.

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A337964 Number of chiral pairs of colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

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A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

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