发表机构
School of Computer Science and Mathematics, Keele University; School of Mathematics and Statistics, The University of Melbourne; Department of Mathematical and Statistical Sciences, Marquette University（基尔大学计算机科学与数学学院; 墨尔本大学数学与统计学院; 马凯特大学数学与统计科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对避免模式的对合枚举研究较少的问题，推导两类4-模式避免对合的Wilf等价类的代数生成函数，调整Mosaic方法扩展其余两类序列初始项并分析其渐近行为。
AI 中文摘要
过去几十年间，避免模式的排列的枚举一直是热门研究领域，但避免模式的对合相关研究却相对较少。本文推导了两类避免长度为4的单个模式的对合的Wilf等价类的代数生成函数，分别为$\boldsymbol{\text{Av}^I}(2431)$和$\boldsymbol{\text{Av}^I}(3421)$。随后，我们将排列的快速计数算法Mosaic方法调整用于对合计数，并将其应用于大幅扩展其余两类避免长度为4的单个模式的Wilf等价类的计数序列的已知初始项，即$\boldsymbol{\text{Av}^I}(1324)$和$\boldsymbol{\text{Av}^I}(4231)$。基于这些扩展序列，我们实证分析了这两类序列的渐近行为。
英文摘要
The enumeration of pattern-avoiding permutations has been a popular area of study over the past several decades, but comparatively little attention has been given to the topic of pattern-avoiding involutions. In this paper, we derive the algebraic generating functions of two Wilf-equivalence classes of involutions avoiding a single pattern of length $4$, $\operatorname{Av^I}(2431)$ and $\operatorname{Av^I}(3421)$. We then adapt the Mosaic method, a fast counting algorithm for permutations, to count involutions and apply it to substantially extend the known initial terms of the counting sequences for the remaining two Wilf-equivalence classes avoiding a pattern of length $4$, $\operatorname{Av^I}(1324)$ and $\operatorname{Av^I}(4231)$. Based on these extended sequences, we empirically analyze the asymptotic behavior of the counting sequences of these two classes.
