On the Enumeration and Asymptotic Analysis of Fibonacci Compositions

TL;DR

This paper analyzes Fibonacci compositions, providing inequalities and asymptotic results, and explores variants with restrictions on consecutive Fibonacci numbers, advancing understanding of their combinatorial properties.

Contribution

It introduces new inequalities, asymptotic analyses, and variants of Fibonacci compositions, expanding the combinatorial understanding of these structures.

Findings

Established inequalities comparing Fibonacci and regular compositions.

Derived asymptotic formulas for Fibonacci compositions.

Explored variants with restrictions on consecutive Fibonacci numbers.

Abstract

We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where summands have a maximum possible value. Then, we consider asymptotic properties of Fibonacci compositions, comparing them to compositions whose terms come from positive linear recurrence sequences. Finally, we consider analogues of these results where we do not allow the use of a certain number of consecutive Fibonacci numbers starting from $F_2 = 1$.