A128766 Number of inequivalent n-colorings of the vertices of the 3D cube under full orthogonal group of the cube (of order 48).

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

Offset: 1

Author

Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007

Keywords

• nonn
• easy

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.

Examples

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

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. 371-384. 2004.
• Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional 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. 63-66.
• Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.

Links

• 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. 371-384. 2004.
• Perez-Aguila, Ricardo, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.
• Perez-Aguila, Ricardo, Enumerating the Configurations in the n-Dimensional 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. 63-66.

Crossrefs

Cf. A000616, A002817.

Cf. A000543 Number when mirror images are counted separately.

Programs

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 *)

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)/(1-x)^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.