Compositions into Powers of $b$: Asymptotic Enumeration and Parameters

TL;DR

This paper analyzes the asymptotic behavior of compositions into powers of a fixed base, providing precise formulas and exploring related parameters using generating functions and transfer matrices.

Contribution

It introduces a novel asymptotic analysis of compositions into powers of $b$, improving previous results with explicit formulas and distribution insights.

Findings

Asymptotic formulas for the number of compositions into powers of $b$

Distribution results for the largest denominator and number of distinct parts

Enhanced understanding of composition enumeration with generating functions

Abstract

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated.