

A128766


Number of inequivalent ncolorings of the vertices of the 3D cube under full orthogonal group of the cube (of order 48).


10



1, 22, 267, 1996, 10375, 41406, 135877, 384112, 966141, 2212750, 4693711, 9340332, 17610307, 31703686, 54839625, 91604416, 148382137, 233880102, 359762131, 541403500, 798782271, 1157522542, 1650105997, 2317268976, 3209603125
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OFFSET

1,2


COMMENTS

The formula was obtained by computing the cycle index of the group of geometric transformations, in 3D space, generated by all possible compositions of the 3 main reflections and the 3 main rotations and their inverses, in any order, with repetition of these geometric transformations allowed.
I assume this refers to colorings of the vertices of the cube.  N. J. A. Sloane, Apr 06 2007
Also the number of ways to color the faces of a regular octahedron with n colors, counting each pair of mirror images as one.


REFERENCES

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371384. 2004.
PerezAguila, Ricardo. Enumerating the Configurations in the nDimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 6366.
Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. SpringerVerlag, 1987.


LINKS



FORMULA

a(n) = (1/48)*(20*n^2 + 21*n^4 + 6*n^6 + n^8).
G.f.: x*(1+x)*(1+12*x+93*x^2+208*x^3+93*x^4+12*x^5+x^6)/(1x)^9. [Colin Barker, Mar 08 2012]
Cycle Index is (1/48)*(s[1]^8 + 6*s[1]^4*s[2]^2 + 13*s[2]^4 + 8*s[1]^2*s[3]^2 + 12*s[4]^2 + 8*s[2]*s[6])  Geoffrey Critzer, Mar 31 2013
a(n)=C(n,1)+20C(n,2)+204C(n,3)+1056C(n,4)+2850C(n,5)+4080C(n,6)+2940C(n,7)+840C(n,8). Each term indicates the number of ways to use n colors to color the cube vertices (octahedron faces) with exactly 1, 2, 3, 4, 5, 6, 7, or 8 colors.


EXAMPLE

a(2)=22 because there are 22 inequivalent 2colorings of the 3D cube, including two for which all of the vertices have the same color.


MATHEMATICA

A[n_] := (1/48)*(20*n^2 + 21*n^4 + 6*n^6 + n^8)
(*or*)
Drop[Table[CycleIndex[GraphData[{"Hypercube", 3}, "Automorphisms"], s]/.Table[s[i]>n, {i, 1, 8}], {n, 0, 25}], 1] (* Geoffrey Critzer, Mar 31 2013 *)


CROSSREFS

Cf. A000543 Number when mirror images are counted separately.


KEYWORD

nonn,easy


AUTHOR

Ricardo PerezAguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007


STATUS

approved



