Computational combinatorics of hyperplane colorings of 6D-hypercube for all irreducible representations and applications

Abstract

Computational generating function techniques are outlined for combinatorics of colorings of all hyperplanes of the 6D-hypercube for 65 irreducible representations of the 6D-hyperoctahedral group isomorphic to the wreath product S6[S2] group of order 46,080. The computational techniques are inspired by a number of physico-chemical and biological applications to molecular chirality, molecular clusters, isomerization reaction graphs, relativistic effects, massively-large data sets, visualization, and genetic regulatory networks. Computational techniques are comprised of computing the generalized character cycle indices of 65 irreducible representations for all hyperplanes of the 6D-hypercube using the Mobius inversion technique followed by the construction of polynomial generators for different cycle types under the hyperoctahedral group action for all six types of hyperplanes of the 6D-hypercube. Subsequently, multinomial generating functions for colorings of all (6-q)-hyperplanes of the 6D-hypercube are constructed for q = 1 through 6. We have presented tables thus computed for the combinatorics for colorings of six hyperplanes of 6D-hypercube for 65 irreducible representations and outline applications to chemical and biological sciences.