A129347 Number of inequivalent n-colorings of the 5D hypercube under the set of geometric transformations generated by all possible compositions of the 5 main reflections and the 10 main rotations and their inverses, in any order, with repetition of these geometric transformations allowed.

1, 1228158, 484086357207, 4805323147589984, 6063609955178082875, 2072592733807533035358, 287612372569381586086269, 20632358601785638477436416, 894188910508179779377279557

Offset

1,2

Comments

The formula was obtained by computing the cycle index of the group of geometric transformations, in 5D space, generated by all possible compositions of the 5 main reflections and the 10 main rotations and their inverses, in any order, with repetition of these geometric transformations allowed. The cycle index was obtained through the well known Polya's Enumeration Theorem.

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 Countings 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

Table of n, a(n) for n=1..9.

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 Countings 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.

Perez-Aguila, Ricardo, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.

Formula

a(n) = (1/3840)*(1184*n^4 + 1624*n^8 + 240*n^10 + 400*n^12 + 311*n^16 + 60*n^20 + 20*n^24 + n^32)

Example

a(2)=1228158 because there are 1228158 inequivalent 2-colorings of the 5D hypercube.

Mathematica

A[n_] := (1/3840)*(1184*n^4 + 1624*n^8 + 240*n^10 + 400*n^12 + 311*n^16 + 60*n^20 + 20*n^24 + n^32)

Crossrefs

Cf. A000616, A002817.

Sequence in context: A203259 A192219 A282424 * A071146 A178477 A144694

Adjacent sequences: A129344 A129345 A129346 * A129348 A129349 A129350

Keyword

nonn,uned

Author

Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 10 2007

Status

approved