多联骨牌密度

中文摘要 AI 辅助

德布鲁因多联骨牌是具有彩色单元格的多联骨牌，且恰好包含另一个多联骨牌用这些颜色进行的所有可能着色的各一个实例，这推广了德布鲁因序列的概念。我们关注最小尺寸的德布鲁因多联骨牌，为寻找这类多联骨牌，我们引入了寻找包含多联骨牌p的至少N个平移副本（允许重叠）的最小多联骨牌的问题，称这类多联骨牌为(p,N)-密集多联骨牌，其尺寸记为a_{p,N}。对于尺寸相同的某些多联骨牌对p和q，我们证明存在多联骨牌变换关联a_{p,N}和a_{q,N}；利用这些变换，我们确定了具有相同序列(a_{p,N})_{N=1}^∞的多联骨牌类，还给出了最多含5个单元格的多数多联骨牌对应的(a_{p,N})_{N=1}^∞的闭式表达式。借助这些结果，我们引入了新型德布鲁因多联骨牌，以及其他具有类德布鲁因性质的彩色多联形。

英文摘要

A de Bruijn polyomino has colored cells and includes exactly one instance of each possible coloring of another polyomino with those colors. This generalizes the notion of a de Bruijn sequence. We are interested in the de Bruijn polyominoes of minimum size. As a step toward finding these, we introduce the problem of finding the smallest polyominoes containing at least $N$ translated copies of the polyomino $p$, allowing overlaps. We say these are $(p,N)$-dense and have size $a_{p,N}$. For certain pairs of polyominoes $p$ and $q$ of equal size, we show there are polyomino transformations relating $a_{p,N}$ and $a_{q,N}$. Using these transformations, we identify classes of polyominoes which share the sequence $(a_{p,N})_{N=1}^\infty$. We give closed forms for $(a_{p,N})_{N=1}^\infty$ for most polyominoes with up to five cells. Leveraging these results we introduce novel de Bruijn polyominoes, as well as other colored polyforms with de Bruijn-like properties.

发表机构

• St. Lawrence University（圣劳伦斯大学）
• Ohio State University（俄亥俄州立大学）
• Georgia Institute of Technology（佐治亚理工学院）

机构由 AI 辅助整理，请以论文原文为准。