Generating Trees and Fibonacci Polyominoes

TL;DR

This paper introduces and analyzes a new class of polyominoes called p-Fibonacci polyominoes, using generating functions to enumerate them and establishing connections with Fibonacci words and integer compositions.

Contribution

It defines p-Fibonacci polyominoes, develops enumeration methods via generating functions, and establishes bijections with Fibonacci words and restricted compositions.

Findings

Derived explicit formulas for enumeration parameters.

Established bijections with Fibonacci words and compositions.

Provided insights into geometric properties of p-Fibonacci polyominoes.

Abstract

We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the number of inner points. Additionally, we establish bijections between Fibonacci polyominoes, binary Fibonacci words, and integer compositions with certain restrictions.