A Comparison of Integer Partitions Based on Smallest Part

TL;DR

This paper studies the generating series related to partitions with bounded differences between largest and smallest parts, proving their eventual positivity for general cases and specific nonnegativity for s=3.

Contribution

It extends previous work by establishing the eventual positivity of the series for all s and providing a detailed nonnegativity result for s=3.

Findings

Proves the eventual positivity of G_{L,s}(q) for all s.

Provides a precise nonnegativity result for the case s=3.

Builds on earlier results focused on s=1 and s=2.

Abstract

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu in 2019. Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general s and also find a precise nonnegativity result for the case $s=3$.