De Bruijn Polyominoes

TL;DR

This paper introduces de Bruijn polyominoes, a generalization of de Bruijn sequences, exploring their structure, construction, and enumeration, with findings suggesting complexity varies with the parity of the polyomino size.

Contribution

It defines de Bruijn polyominoes and prismatic polyominoes, analyzes their shape and coloring, and investigates how their properties depend on the size parity of the base polyomino.

Findings

Shape and construction methods for prismatic polyominoes.

Enumeration techniques for colorings of prismatic polyominoes.

Evidence that problem difficulty depends on the parity of polyomino size.

Abstract

We introduce the notions of de Bruijn polyominoes and prismatic polyominoes, which generalize the notions of de Bruijn sequences and arrays. Given a small fixed polyomino $p$ and a set of colors $[n]$, a de Bruijn polyomino for $(p,n)$ is a colored fixed polyomino $P$ with cells colored from $[n]$ such that every possible coloring of $p$ from $[n]$ exists as a subset of $P$. We call de Bruijn polyominoes for $(p,n)$ of minimum size $(p,n)$-prismatic. We discuss for some values of $p$ and $n$ the shape of a $(p,n)$-prismatic polyomino $P$, the construction of a coloring of $P$, and the enumeration of the colorings of $P$. We find evidence that the difficulty of these problems may depend on the parity of the size of $p$