关于模式替换等价的新结果：一个经典定理的推广和一个新猜想的修正
New Results on Pattern-Replacement Equivalences: Generalizing a Classical Theorem and Revising a Recent Conjecture
摘要：
本文研究了长度为$n$的置换集$S_n$上的模式替换等价关系。每个等价关系由一组模式确定，等价排列通过模式替换以类似于Knuth关系的方式连接。我们的一个主要结果是将著名的Erdos-Szekeres置换模式回避定理推广到置换模式替换的一个新结果。特别是，我们表明，在$\{123 \cdots k, k \cdots 321\}$ -等价，$S_n$中的所有排列是等价的奇偶校验时，$n \ge Ω(k^2)$。此外，我们扩展了Kuszmaul和周的工作上的一个无限的家庭的模式替换等价称为旋转等价。Kuszmaul和Zhou证明了在$S_n$中旋转等价总是产生一个或两个非平凡等价类，并且证明了非平凡类的数量只取决于旋转等价中涉及的模式（而不是$n$）。我们提出了一个反例，他们的猜想，并证明了一个新的定理完全分类（大$n$）时，有一个非平凡的等价类，当有两个非平凡的等价类。最后，我们计算分析的模式替换等价的长度为4的模式对集。然后，我们专注于三种情况下，其中的非平凡等价类的数量匹配OEIS序列。对于其中两个，我们提出了充分的证据的枚举和第三，我们建议一个潜在的未来的证明方法。
英文摘要：
In this paper we study pattern-replacement equivalence relations on the set $S_n$ of permutations of length $n$. Each equivalence relation is determined by a set of patterns, and equivalent permutations are connected by pattern-replacements in a manner similar to that of the Knuth relation. One of our main results generalizes the celebrated Erdos-Szekeres Theorem for permutation pattern-avoidance to a new result for permutation pattern-replacement. In particular, we show that under the $\{123 \cdots k, k \cdots 321\}$-equivalence, all permutations in $S_n$ are equivalent up to parity when $n \ge Ω(k^2)$. Additionally, we extend the work of Kuszmaul and Zhou on an infinite family of pattern-replacement equivalences known as the rotational equivalences. Kuszmaul and Zhou proved that the rotational equivalences always yield either one or two nontrivial equivalence classes in $S_n$, and conjectured that the number of nontrivial classes depended only on the patterns involved in the rotational equivalence (rather than on $n$). We present a counterexample to their conjecture, and prove a new theorem fully classifying (for large $n$) when there is one nontrivial equivalence class and when there are two nontrivial equivalence classes. Finally, we computationally analyze the pattern-replacement equivalences given by sets of pairs of patterns of length four. We then focus on three cases, in which the number of nontrivial equivalence classes matches an OEIS sequence. For two of these we present full proofs of the enumeration and for the third we suggest a potential future method of proof.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
