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.
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
KEYWORD
nonn,easy
AUTHOR
Clint. C. Williams (Clintwill(AT)aol.com)
EXTENSIONS
Entry revised by N. J. A. Sloane, Jan 03 2005
STATUS
approved