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

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

0,3

Comments

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)

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

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).

Formula

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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).

Sequence in context: A022747 A270498 A260727 * A220648 A243422 A028110

Adjacent sequences: A000540 A000541 A000542 * A000544 A000545 A000546

Keyword

nonn,easy

Author

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

Extensions

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

Status

approved