论文标题
基于最小部分的整数分区的比较
A Comparison of Integer Partitions Based on Smallest Part
论文作者
论文摘要
对于正整数$ n，l $和$ s $，请考虑以下两套包含$ n $的分区，其最大和最小零件之间的差异为$ l $：第一组包含最小零件$ s $的分区，而第二组包含至少$ s+1+1 $的分区。令$ g_ {l，s}（q）$为生成系列，其系数为$ q^n $是上述两组分区的大小之间的差异。该生成系列是由Berkovich和UNCU在2019年推出的。先前的结果集中在$ g_ {l，s}（q）$的非负性上，$ s = 1 $和$ s = 2 $。在本文中，我们显示了一般S的$ g_ {l，s}（q）$的最终阳性，并且还为情况找到了确切的非阴性结果$ s = 3 $。
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 $s=3$.
