Counting Polyominoes in a Rectangle b x h

TL;DR

This paper introduces automata-based methods to count polyominoes within fixed-width rectangles, enabling the derivation of generating functions for small widths and increasing heights.

Contribution

It presents a novel automata approach to systematically generate and analyze polyominoes inscribed in rectangles of fixed width, advancing combinatorial enumeration techniques.

Findings

Automata can generate polyominoes in fixed-width rectangles.

Generating functions are obtained for small widths.

The method facilitates enumeration of polyominoes with increasing height.

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.