Exhaustive generation of `Mrs Perkins's quilt' square dissections for low orders

TL;DR

This paper presents a graph-based method for exhaustively generating all Mrs Perkins's quilt square dissections up to order 18, confirming and extending known integer sequences through computational enumeration.

Contribution

It introduces a graph representation for dissections and uses plantri software for exhaustive generation, extending previous results to higher orders.

Findings

Generated all dissections up to order 18

Confirmed and extended existing integer sequences

Validated results with direct enumeration

Abstract

Dissections of a square into smaller squares, with the smaller squares having relatively prime sizes, are known as Mrs Perkins's quilts. A representation of these dissections using graphs is presented. The edges are directed and coloured North-South or West-East, and the graph corresponds naturally to the dissection. This representation allowed the exhaustive generation of all dissections up to order 18, using the plantri software. The results were cross-checked by generating all dissections of small sizes using a direct approach. The results confirm, extend and introduce several integer sequences.