Combinatorial Identities Via Phi Functions and Relatively Prime Subsets

TL;DR

This paper develops combinatorial formulas involving phi functions and Mertens function to count subsets of positive integers with specific coprimality properties, extending previous results for special sets.

Contribution

It introduces new formulas for counting relatively prime subsets and elements, generalizing prior work to arbitrary finite sets of positive integers.

Findings

Formulas for counting subsets relatively prime to n

Characterizations of sets with pairwise coprime elements

Connections to Mertens function in combinatorial formulas

Abstract

Let $n$ be a positive integer and let $A$ be nonempty finite set of positive integers. We say that $A$ is relatively prime if $\gcd(A) =1$ and that $A$ is relatively prime to $n$ if $\gcd(A,n)=1$. In this work we count the number of nonempty subsets of $A$ which are relatively prime and the number of nonempty subsets of $A$ which are relatively prime to $n$. Related formulas are also obtained for the number of such subsets having some fixed cardinality. This extends previous work for the cases where $A$ is an interval or a set in arithmetic progression. Applications include: a) An exact formula is obtained for the number of elements of $A$ which are co-prime to $n$; note that this number is $\phi(n)$ if $A=[1,n]$. b) Algebraic characterizations are found for a nonempty finite set of positive integers to have elements which are all pairwise co-prime and consequently a formula is given for the number of nonempty subsets of $A$ whose elements are pairwise co-prime. c) We provide combinatorial formulas involving Mertens function.