论文标题
关于不允许的集合和完美的功率频率的Berkovich-Uncu型分区不平等
Berkovich-Uncu type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies
论文作者
论文摘要
最近，藤和第一作者（Ann。Comb。25（2021）697-728）证明了伯科维奇和UNCU的猜想不平等（Ann。Comb。23（2019）263-284），内容涉及与无效的部分。在本文中，我们考虑了不允许的部分，从而概括了这种不平等。我们将它们与分区进行比较，其某些部分的频率是完美的t^{th}功率。我们的不等式在一定结合后成立，对于给定的t是s的多项式，对t = 1的先前已知结合的重大改善。为了证明这些不平等现象，我们的方法涉及在相关分区集之间构建注射图。这些地图的构造至关重要地涉及分析和演算的概念，例如用于证明N^T的明确地图，以及Jensen对凸功能的不平等，然后将它们与数字理论（例如Frobenius数字，一致性类别，一致性类别，二进制数字和Quadratic残留物）合并。我们还显示了结果与彩色分区的联系。最后，我们提出了一个开放的问题，似乎与功率残基和对角三元二次形式的几乎普遍性有关。
Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of N^t, and Jensen's inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.
