Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications

Abstract

We obtain the generating functions for the combinatorial enumeration of colorings of all hyperplanes of hypercubes for all irreducible representations of the hyperoctahedral groups. The computational group theoretical techniques involve the construction of generalized character cycle indices of all irreducible representations for all hyperplanes of the hypercube using the M\"{o}bius function, polynomial generators for all cycle types and for all hyperplanes. This is followed by the construction of the generating functions for colorings of all (n-q)-hyperplanes of the hypercube, for example, vertices (q=5), edges (q=4), faces (q=3), cells (q=2) and tesseracts (q=4) for a 5D-hypercube. Tables are constructed for the combinatorial numbers for coloring all hyperplanes of 5D-hypercubes for 36 irreducible representations. Applications to chirality, chemistry and biology are also pointed out.