Bin Decompositions

TL;DR

This paper introduces the $(n,m)$-bin sequence, a new class of sequences for representing positive integers uniquely, and demonstrates that the number of summands in their decompositions follows a Gaussian distribution.

Contribution

It defines the $(n,m)$-bin sequence, proves the existence and uniqueness of legal decompositions, and shows their summand count distribution is Gaussian.

Findings

Legal decompositions are unique for the $(n,m)$-bin sequence.

Number of summands in decompositions follows a Gaussian distribution.

Sequences are not always positive linear recurrences but still have similar properties.

Abstract

It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n =F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any $n,m\in\mathbb{N}$ we create a sequence called the $(n,m)$-bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the $(n,m)$-bin legal decompositions displays Gaussian behavior.