On the lower bound for the number ofn-ominoes

Joseph Migga Kizza, University of Nebraska - Lincoln

Abstract

Unit squares having their vertices at integer points in the Cartesian plane are called cells. A finite set of integer points forming a vertex set of a connected subgraph of the square-lattice-graph G in the Cartesian plane is called a polyomino. A polyomino with exactly n cells is called an n-omino. Two n-ominoes are considered the same if there is an isometry which maps one onto the other. Let (r,s) be fixed integer points in $Z \times Z$ and define $\tau\sb{r,s}$: $Z \times Z \to\ Z \times Z$ as $\tau\sb{r,s}(x,y) = (x + r,y + s)$ for all (x,y) $\in\ Z \times Z$. The set T = $\{(r,s)$: $(r,s) \in Z \times Z\}$ is a translation group. Two subsets of $Z \times Z$, say P and Q are said to be translationally equivalent just when there exists $\tau\sb{r,s} \in T$ such that P = $\{\tau\sb{r,s}(x,y)$: $(x,y) \in Q\}$. If P and Q are two finite sets of points corresponding to polyominoes, and P and Q are translationally equivalent, we say the polyominoes are translations. Let t(n) be the number of all translation equivalence classes of n-ominoes. It is known that $lim\sb{n\to\infty}(t(n))\sp{1/n}=\theta$ exists and this limit is known as the Klarner Constant. The value of $\theta$ itself is not known. Let $t\sb{k}(n)$ denote the number of all translational type n-ominoes that fit in a strip of width k. For all k and n, $t\sb{k}(n)\leq t(n)$. One can show that $lim\sb{n\to\infty}(t\sb{k}(n))\sp{1/n}=\tau\sb{k}$ exists also. Hence $\tau\sb{k}\leq\theta$ for all k. We will show that $lim\sb{n\to\infty}(t\sb{k}(n))\sp{1/n}$ tends to a limit $\tau\sb{k}$. Also that $\tau\sb{k}\leq\tau\sb{k+1}$.

Subject Area

Computer science|Mathematics

Recommended Citation

Kizza, Joseph Migga, "On the lower bound for the number ofn-ominoes" (1990). ETD collection for University of Nebraska-Lincoln. AAI9108227.
https://digitalcommons.unl.edu/dissertations/AAI9108227

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