De Bruijn Polyominoes

Abstract

We introduce the notion of de Bruijn polyominoes, which generalizes the notions of de Bruijn sequences and arrays. Given a polyomino $p$ and a positive integer $n$, a $(p,n)$-de Bruijn polyomino is a colored polyomino $P$ with cells colored from $\{1,...,n\}$ such that every possible coloring of $p$ from $\{1,...,n\}$ occurs within $P$, not counting rotations. We discuss for some polyominoes $p$ and integers $n$ the shape of the smallest $(p,n)$-de Bruijn polyomino $P$, the construction of a coloring of $P$, and the enumeration of the de Bruijn colorings of $P$.