Recursive Symmetries: Chemically Induced Combinatorics of Colorings of Hyperplanes of an 8-Cube for All Irreducible Representations

Abstract

We outline symmetry-based combinatorial and computational techniques to enumerate the colorings of all the hyperplanes (q = 1–8) of the 8-dimensional hypercube (8-cube) and for all 185 irreducible representations (IRs) of the 8-dimensional hyperoctahedral group, which contains 10,321,920 symmetry operations. The combinatorial techniques invoke the Möbius inversion method in conjunction with the generalized character cycle indices for all 185 IRs to obtain the generating functions for the colorings of eight kinds of hyperplanes of the 8-cube, such as vertices, edges, faces, cells, tesseracts, and hepteracts. We provide the computed tables for the colorings of all the hyperplanes of the 8-cube. We also show that the developed techniques have a number of chemical, biological, chiral, and other applications that make use of such recursive symmetries.