Arrow-Wilf 等价性与短箭头模式的枚举结果

中文摘要 AI 辅助

箭头模式由 Berman 和 Tenner 引入，为研究同时包含单行和循环结构约束的排列类提供了一个统一框架。在本文中，我们继续 Archer 和 Laudone 发起的对箭头模式避免的系统性研究。我们建立了若干结构结果，包括一个关键引理，该引理在特定条件下将箭头模式转化为文氏模式，并推导出一系列由反转、补集和插入操作产生的 Arrow-Wilf 等价关系。我们还解决了 Archer 和 Laudone 留下的两个未决情形 $(12;3\to 3)$ 和 $(21;3\to 3)$，并枚举了形式为 $(\nu; b\to c)$ 且大小为 $3$ 的箭头模式，其中 $\nu \in \{31, 23, 32\}$，$b,c\in [3]$，提供了将结果与贝尔数、贝塞尔数、卡特兰数和错排数联系起来的显式公式。结合 Archer 和 Laudone 的早期工作，对于 $|\nu|\le 2$，仅剩 $(32;1\to 3)$ 未解决，我们将其作为开放问题提出。

英文摘要

Arrow patterns, introduced by Berman and Tenner, provide a unified framework for studying permutation classes where both one-line and cycle structure constraints are present. In this paper, we continue the systematic study of arrow pattern avoidance initiated by Archer and Laudone. We establish several structural results, including a key lemma that translates arrow patterns into vincular patterns under certain conditions, and derive a series of arrow-Wilf equivalences arising from reversal, complementation, and insertion operations. We also resolve the two cases $(12;3\to 3)$ and $(21;3\to 3)$ left open by Archer and Laudone, and enumerate the arrow patterns of the form $(ν; b\to c)$ of size $3$ with $ν\in \{31, 23, 32\}$ and $b,c\in [3]$, providing explicit formulas connecting the results to Bell numbers, Bessel numbers, Catalan numbers, and derangement numbers. Together with earlier work of Archer and Laudone, this leaves only $(32;1\to 3)$ unresolved for $|ν|\le 2$, which we pose as an open problem.

发表机构

• College of Mathematics Physics and Information Shaoxing University（绍兴学院数学与信息技术学院）

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