Stack-sorting, set partitions, and Lassalle's sequence

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:175

Article

Defant, Colin¹  Engen, Michael²  Miller, Jordan A.³ 

[1] Princeton Univ, Princeton, NJ 08544 USA

[2] Univ Florida, Gainesville, FL 32611 USA

[3] Washington State Univ, Pullman, WA 99164 USA

关键词: Stack-sorting;    Set partition;    Lassalle's sequence;    Cumulant;    Uniquely sorted permutation;    Tutte polynomial;    Valid hook configuration;   

DOI  :  10.1016/j.jcta.2020.105275

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free