Enumerating and identifying semiperfect colorings of symmetrical patterns

TL;DR

This paper develops methods to identify and count all inequivalent semiperfect colorings of symmetrical patterns, focusing on colorings where the color group is a subgroup of the pattern's symmetry group with index 2.

Contribution

It introduces a novel approach to enumerate and classify semiperfect colorings using group partitions and coset representatives, establishing a correspondence with conjugate subgroups.

Findings

Provides explicit enumeration techniques for semiperfect colorings.

Establishes a one-to-one correspondence between colorings and conjugate subgroups.

Offers a framework for analyzing symmetry-based colorings in patterns.

Abstract

If $G$ is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group $H$ is a subgroup of $G$ of index 2. We give results on how to identify and enumerate all inequivalent semiperfect colorings of certain patterns. This is achieved by treating a coloring as a partition $\{hJ_iY_i:i\in I,h\in H\}$ of $G$, where $H$ is a subgroup of index 2 in $G$, $J_i\leq H$ for $i\in I$, and $Y=\cup_{i\in I}{Y_i}$ is a complete set of right coset representatives of $H$ in $G$. We also give a one-to-one correspondence between inequivalent semiperfect colorings whose associated color groups are conjugate subgroups with respect to the normalizer of $G$ in the group of isometries of $\mathbf{R}^n$.