当箭头模式与经典模式相遇时
When arrow patterns meet classical patterns
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- Chongqing University（重庆大学）
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中文总结 AI 辅助
研究旨在弥合排列的循环表示法和单行表示法之间的结构鸿沟，通过引入箭头模式概念，解决了Archer和Laudone留下的三个关于箭头模式避免的猜想，包括枚举特定子类排列及给出不同证明。
中文摘要 AI 辅助
为弥合排列的循环表示法与其单行表示法之间的结构鸿沟，Berman和Tenner引入了一种称为箭头模式的排列模式新定义。最近，Archer和Laudone开启了对箭头模式避免的系统研究，留下了三个有趣的猜想。在本文中，我们解决了所有这三个猜想。首先，我们枚举了同时避免长度为3的经典模式和固定长度为3的箭头模式的排列的所有六个子类，从而证实了前两个猜想。其次，我们通过提供两个独立的证明解决了第三个猜想（它涉及不同的箭头模式）。这些证明分别依赖于Biane双射对非嵌套对合的限制以及Krattenthaler从避免321的排列到Dyck路径的双射。
英文摘要
Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively.
