A Comparison of Integer Partitions Based on Smallest Part

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 (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 $s=3$.