

A000543


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


14



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
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. 144184 (see p. 147).


LINKS



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)/(1x)^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.


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)/(1x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 15 2012 *)


PROG



CROSSREFS

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).


KEYWORD

nonn,easy


AUTHOR

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


EXTENSIONS



STATUS

approved



