Gap between the largest and smallest parts of partitions and Berkovich and Uncu's conjectures

Wenston J. T. Zang, Jiang Zeng

备注：: 12 pages

英文摘要：

We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our theorems are stronger than their original conjectures. The analytic version of our results shows that the coefficients of some partition $q$-series are eventually positive.

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