

A129347


Number of inequivalent ncolorings 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.


0



1, 1228158, 484086357207, 4805323147589984, 6063609955178082875, 2072592733807533035358, 287612372569381586086269, 20632358601785638477436416, 894188910508179779377279557
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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. 371384, 2004.
PerezAguila, Ricardo. Enumerating the Configurations in the nDimensional 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. 6366.
Polya, G. & Read, R. C., Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. SpringerVerlag, 1987.


LINKS



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 2colorings 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



KEYWORD

nonn,uned


AUTHOR

Ricardo PerezAguila (ricardo.perez.aguila(AT)gmail.com), Apr 10 2007


STATUS

approved



