On the number of even and odd strings along the over partitions of n

TL;DR

This paper proves that for large integers, the number of odd strings exceeds or equals the number of even strings in overpartitions, introduces m-strings, and confirms related positivity conjectures.

Contribution

It establishes the inequality between odd and even strings in overpartitions for large n, introduces m-strings, and confirms a positivity conjecture.

Findings

A_k(n) ≥ B_k(n) for large n

Introduction of m-strings and their relation to positivity conjectures

Confirmation of the positivity conjecture for large n

Abstract

Recently, Andrews, Chan, Kim and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture $A_k (n) \geq B_k (n)$ holds for large enough positive integers n, where A_k(n) (resp. B_k(n)) is the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show that how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.