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

0,3

Comments

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)

References

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).

Links

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).

Formula

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)

Mathematica

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

Prog

(PARI) { 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

Crossrefs

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).

Sequence in context: A224741 A209824 A230333 * A252997 A126829 A171406

Adjacent sequences: A060527 A060528 A060529 * A060531 A060532 A060533

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 11 2001

Extensions

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

Status

approved