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Counting Polyominoes in a Rectangle b x h

Louis Marin

TL;DR

Methods to automatically obtain automata that generate polyominoes inscribed in a rectangle of fixed width and increasing height are provided and used to obtain the generating function of those sequences for small widths.

Abstract

In this paper, we provide methods to automatically obtain automata that generate polyominoes inscribed in a rectangle of fixed width and increasing height. We use them to obtain the generating function of those sequences for small widths.

Counting Polyominoes in a Rectangle b x h

TL;DR

Methods to automatically obtain automata that generate polyominoes inscribed in a rectangle of fixed width and increasing height are provided and used to obtain the generating function of those sequences for small widths.

Abstract

In this paper, we provide methods to automatically obtain automata that generate polyominoes inscribed in a rectangle of fixed width and increasing height. We use them to obtain the generating function of those sequences for small widths.

Paper Structure

This paper contains 3 sections, 1 theorem, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Building $\mathcal{A}_b$
  3. Generating the automaton $\mathcal{A}_b$.

Key Result

Theorem 1

The set of stacks of size $h$ of words recognized by $\mathcal{A}_b$ is in bijection with the set of polyominoes inscribed in a rectangle of size $b \times h$.

Figures (2)

  • Figure 1: Example of a word recognized by $\mathcal{A}_5$ and the associated polyomino
  • Figure 2: Number of states in $\mathcal{A}_b$ for $b = 0, \dots,11$

Theorems & Definitions (2)

  • Example 1
  • Theorem 1