AI 中文总结
本文研究同时避免长度为3的经典模式与平坦偏序模式的排列的Wilf等价性，解决Qiu和Remmel的相关猜想并纠正其论文错误，得出不同ℓ对应的Wilf等价类数量。
AI 中文摘要
众所周知，对于每个长度为3的经典模式τ，长度为n的避免τ模式的排列数是第n个卡特兰数，并且已构建并研究了不同长度为3的避免类之间的大量双射。在本文中，我们通过研究同时避免一个长度为3的经典模式和一个平坦偏序模式（POP）的排列之间的Wilf等价性，来细化这个经典问题。偏序模式（POP）为编码经典排列模式族提供了一个灵活的框架。对于ℓ≥3且1≤x≤ℓ，设P_{ℓ,x}为长度为ℓ的POP，其中位置x处的元素必须小于所有其他元素，而其余元素之间无任何关系，这类POP被称为平坦POP。我们对所有对(τ,P_{ℓ,x})之间的Wilf等价性进行分类，其中τ是长度为3的经典模式。对于每个ℓ≥4，所得的6ℓ个对恰好构成2ℓ−1个Wilf等价类，而特殊情况ℓ=3给出4个类。我们的证明结合了显式公式和递推关系的推导与双射的构建。此外，我们引入了新颖的素因子论证来区分剩余的候选类，将问题简化为证明对于ℓ≥3274，某个丢番图方程无解，其中3274这个边界不被声称是尖锐的。最后，通过扩展我们关于POP的工作，我们解决了Qiu和Remmel关于132-避免排列上象限标记网格模式分布的一个猜想，并纠正了他们论文中对证明至关重要的一个错误。
英文摘要
It is well known that, for each classical pattern $τ$ of length 3, the number of $τ$-avoiding permutations of length $n$ is the $n$th Catalan number, and numerous bijections between different length-three avoidance classes have been constructed and studied. In this paper, we refine this classical problem by studying Wilf equivalence among permutations that simultaneously avoid a classical pattern of length three and a flat partially ordered pattern. Partially ordered patterns (POPs) provide a flexible framework for encoding families of classical permutation patterns. For $\ell\geq 3$ and $1\leq x\leq\ell$, let $P_{\ell,x}$ be the length-$\ell$ POP in which the entry at position $x$ is required to be smaller than all the other entries, while no relations are imposed among the remaining entries. Such POPs are called flat POPs. We classify the Wilf equivalences among all pairs $(τ,P_{\ell,x})$, where $τ$ is a classical pattern of length three. For every $\ell\geq4$, the resulting $6\ell$ pairs form exactly $2\ell-1$ Wilf equivalence classes, while the exceptional case $\ell=3$ gives four classes. Our proofs combine the derivation of explicit formulas and recurrence relations with the construction of bijections. Moreover, we introduce novel prime-divisor arguments to distinguish the remaining candidate classes, reducing the problem to showing that a certain Diophantine equation has no solutions for $\ell\ge 3{,}274$, where the bound $3{,}274$ is not claimed to be sharp. Finally, by extending our work on POPs, we resolve a conjecture of Qiu and Remmel concerning the distribution of quadrant marked mesh patterns on 132-avoiding permutations and correct an error in their paper that is crucial to the proof.
