发表机构
Indian Institute of Technology, Bombay; Indian Institute of Science, Bengaluru（印度孟买理工学院; 印度班加罗尔印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完成了长度为三的模式四元组和五元组的shape-Wilf等价分类，引入Dyck路径方法处理Ferrers板，获得新等价关系、计数公式及匹配枚举的双射解释。
AI 中文摘要
基于我们对长度为三的模式三元组的shape-Wilf等价类的分类，我们完成了四元组和五元组的分类。更大的模式集合展现出未被用于三元组的编码方法所捕捉的结构特征，需要额外的组合工具。我们的主要新工具是一种针对包含阶梯板的Ferrers板的Dyck路径方法。通过引入区分单元及相关结构参数到关联的Dyck路径上，我们获得了新的shape-Wilf等价性，并且对若干类，给出了在固定Ferrers板上避免横截数的显式公式。我们进一步通过Bloom和Elizalde的模式保持双射研究相应的避免模式的完美匹配。除了推导递推关系，我们还以grand Dyck路径、Schröder路径和Dyck路径给出了若干匹配枚举序列的双射解释。这些构造提供了超越分类本身的额外组合结构。连同关于三元组的姊妹篇，本工作完成了长度为三的模式子集的shape-Wilf等价类的分类，除平凡的空集和全集模式集合外。
英文摘要
Building on our classification of shape-Wilf-equivalence classes for triples of patterns of length three, we complete the classification for quadruples and quintuples. The larger pattern sets exhibit structural features that are not captured by the encoding methods used for triples and require additional combinatorial tools. Our main new ingredient is a Dyck-path approach to Ferrers boards containing the staircase board. By introducing distinguished cells and related structural parameters on the associated Dyck paths, we obtain new shape-Wilf-equivalences and, for several classes, explicit formulas for the number of avoiding transversals on a fixed Ferrers board. We further study the corresponding pattern-avoiding perfect matchings through the pattern-preserving bijection of Bloom and Elizalde. Besides deriving recurrence relations, we give bijective interpretations of several matching enumeration sequences in terms of grand Dyck paths, Schröder paths, and Dyck paths. These constructions provide additional combinatorial structure beyond the classification itself. Together with the companion paper on triples, the present work completes the classification of shape-Wilf-equivalence classes for subsets of patterns of length three, apart from the trivial empty and full pattern sets.
