Enumeration of Symmetry Classes of Parallelogram Polyominoes

TL;DR

This paper extends the enumeration of symmetry classes from convex polyominoes to parallelogram polyominoes using subgroup actions, providing new combinatorial insights into their symmetry classifications.

Contribution

It introduces a novel enumeration of symmetry classes of parallelogram polyominoes using subgroup D2 of D4, expanding previous work on convex polyominoes.

Findings

Enumeration of symmetry classes for parallelogram polyominoes

Application of subgroup D2 to classify symmetries

Extension of previous convex polyomino results

Abstract

Parallelogram polyominoes are a subclass of convex polyominoes in the square lattice that has been studied extensively in the literature. Recently congruence classes of convex polyominoes with respect to rotations and reflections have been enumerated by counting orbits under the action of the dihedral group D4, of symmetries of the square, on (translation-type) convex polyominoes. Asymmetric convex polyominoes were also enumerated using Moebius inversion in the lattice of subgroups of D4. Here we extend these results to the subclass of parallelogram polyominos using a subgroup D2 of D4 which acts of this class.