Compositions of n Satisfying Some Coprimality Conditions

TL;DR

This paper derives asymptotic formulas for counting k-compositions of n under specific coprimality constraints, advancing understanding of their distribution and properties.

Contribution

It provides the first exact asymptotic formula for compositions with the first summand coprime to the rest and estimates for compositions with pairwise coprime summands.

Findings

Exact asymptotic count for compositions with first summand coprime to others

Estimate for compositions with all pairwise coprime summands

Enhanced understanding of coprimality conditions in compositions

Abstract

A k-composition of n is a sequence of length k of positive integers summing up to n. In this paper, we investigate the number of k-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic formula for the number of k-compositions having the first summand coprime to the others. Then, we estimate the number of k-compositions whose summands are all pairwise coprime.