Coprime partitions and Jordan totient functions

TL;DR

This paper investigates the relationship between coprime partitions, compositions, and Jordan totient functions, revealing limitations and special cases where these functions can express the counts of such partitions.

Contribution

It establishes that coprime compositions relate linearly to Jordan totient functions, while coprime partitions do not, except in specific cases for large n and small k, and introduces generalized Jordan totient functions.

Findings

Coprime compositions can be expressed as rational linear combinations of Jordan totient functions.

Coprime partitions cannot generally be expressed as rational linear combinations of Jordan totient functions.

For large n, coprime partitions into 2 or 3 parts can be expressed as complex linear combinations of Jordan totient functions.

Abstract

We show that while the number of coprime compositions of a positive integer $n$ into $k$ parts can be expressed as a $\mathbb{Q}$-linear combinations of the Jordan totient functions, this is never possible for the coprime partitions of $n$ into $k$ parts. We also show that the number $p_k'(n)$ of coprime partitions of $n$ into $k$ parts can be expressed as a $\mathbb{C}$-linear combinations of the Jordan totient functions, for $n$ sufficiently large, if and only if $k\in \{2,3\}$ and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that $p_k'(n)$ can be always expressed as a $\mathbb{C}$-linear combinations of them.