For positive integers \(L \ge 3\) and s, Berkovich and Uncu (Ann Comb 23:263–284, 2019) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval \(\{s, \ldots , L+s\}\). Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases \(s=1\) and \(s=2\). In the present article, we prove their conjecture for general s by proving a stronger theorem. We also prove other related conjectures found in the same paper.

中文翻译：

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关于整数分区的最小部分和缺失部分的猜想

对于正整数\(L \ge 3\)和s，Berkovich 和 Uncu (Ann Comb 23:263–284, 2019) 推测了两个密切相关的分区集的大小之间的不等式，这些分区的部分位于区间\(\ {s, \ldots , L+s\}\)。通过指定不允许的部分以及最小部分，对集合施加了进一步的限制。作者在\(s=1\)和\(s=2\)的情况下证明了他们的猜想。在本文中，我们通过证明一个更强的定理来证明他们对一般s的猜想。我们还证明了在同一篇论文中发现的其他相关猜想。