A047780 Number of inequivalent ways to color faces of a cube using at most n colors.

0, 1, 10, 57, 240, 800, 2226, 5390, 11712, 23355, 43450, 76351, 127920, 205842, 319970, 482700, 709376, 1018725, 1433322, 1980085, 2690800, 3602676, 4758930, 6209402, 8011200, 10229375, 12937626, 16219035, 20166832, 24885190, 30490050

Offset: 0

Author

Jud McCranie

Keywords

• nonn
• easy

Comments

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the faces has cycle index (x1^6 + 3*x1^2*x2^2 + 6*x1^2*x4 + 6*x2^3 + 8*x3^2)/24.

a(n) is also the number of inequivalent colorings of the vertices 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 cube face (octahedron vertex) 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_1^6

Vertex rotation 8 x_3^2

Edge rotation 6 x_2^3

Small face rotation 6 x_1^2x_4^1

Large face rotation 3 x_1^2x_2^2 (End)

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 254 (corrected).
• 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).
• M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246 (the formula given is incorrect but was corrected in the second printing).
• J.-P. Delahaye, 'Le miraculeux "lemme de Burnside"','Le coloriage du cube' p. 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• 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 (7,-21,35,-35,21,-7,1)

Crossrefs

Cf. A198833 (unoriented), A093566(n+1) (chiral), A337898 (achiral).

Other elements: A060530 (edges), A000543 (cube vertices, octahedron faces).

Cf. A006008 (tetrahedron), A000545 (dodecahedron faces, icosahedron vertices), A054472 (icosahedron faces, dodecahedron vertices).

Row 3 of A325004 (orthoplex vertices, orthotope facets) and A337887 (orthotope faces, orthoplex peaks).

Programs

Magma

[(n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24: n in [1..30]]; // Vincenzo Librandi, Apr 27 2012

Mathematica

CoefficientList[Series[x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1-x)^7,{x,0,33}],x] (* Vincenzo Librandi, Apr 27 2012 *)

Formula

a(n) = (n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24 = n+8*C(n, 2)+30*C(n, 3)+68*C(n, 4)+75*C(n, 5)+30*C(n, 6). Each term of the RHS indicates the number of ways to use n colors to color the cube faces (octahedron vertices) with exactly 1, 2, 3, 4, 5, or 6 colors.

G.f.: x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1-x)^7. - Colin Barker, Jan 29 2012

a(n) = A198833(n) + A093566(n+1) = 2*A198833(n) - A337898(n) = 2*A093566(n+1) + A337898(n). - Robert A. Russell, Oct 08 2020

Extensions

Corrected version of A006550 and A006529.

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