Toward a Canonical Representation of Blocked Rectangular Grids with an Application to Finite Tiling Problems

TL;DR

This paper develops a canonical representation for blocked rectangular grids to efficiently identify unique configurations under symmetry, aiding in solving finite tiling problems by reducing redundancy.

Contribution

It introduces a method to select representative grids from each symmetry class, minimizing redundancy and enabling complete enumeration of solutions in tiling problems.

Findings

Successfully reduces the number of grids for analysis by eliminating symmetric duplicates.

Provides exact methods for complete canonical representation in specific cases.

Identifies all solvable grids within certain polyomino tiling collections.

Abstract

Given the collection of all $m\times n$ rectangular grids which have a fixed number $1\leq r\leq mn$ of blocked cells, we explicitly describe a proper subset of the collection which is guaranteed to contain at least one grid from each equivalence class under symmetry, eliminating the majority of redundant grids. We analyze the extent to which redundant grids remain in the reduced set, and give general cases in which our methods exactly produce a complete set of canonical representatives for the equivalence classes. As an application of our results, we specify collections of polyomino tiling problems and find all solvable grids in each collection.