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Stack-sorting, set partitions, and Lassalle's sequence
Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2020-05-25 , DOI: 10.1016/j.jcta.2020.105275
Colin Defant , Michael Engen , Jordan A. Miller

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting map. We prove that the sequences obtained by counting these permutations according to their first entries are symmetric, and we conjecture that they are log-concave. We also obtain new recurrence relations involving Lassalle's sequence and the sequence that enumerates valid hook configurations. We end with several suggestions for future work.



中文翻译:

堆栈排序,设置分区和Lassalle的序列

我们展示了最近引入的称为有效钩子配置的组合对象和某些加权集分区之间的双射。将注意力集中在匹配的分区上时,我们获得了Lassalle序列的三种新的组合解释。这些解释之一涉及在(West)堆栈排序图下恰好具有一个原像的置换。我们证明通过根据它们的第一个条目对这些排列进行计数而获得的序列是对称的,并且我们推测它们是对数凹形的。我们还获得了涉及Lassalle序列和枚举有效钩子配置的序列的新递归关系。最后,我们为以后的工作提出了一些建议。

更新日期:2020-05-25
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