De Bruijn Polyominoes

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 $p$