$\{1234,3412\}$ -模式置换下$S_n$的非平凡等价类的计数
Counting the Nontrivial Equivalence Classes of $S_n$ under $\{1234,3412\}$-Pattern-Replacement
摘要：
研究了长度为$n$的置换集$S_n$上的$\{1234, 3412\}$模式替换等价关系，它在概念上类似于Knuth关系.特别是，我们列举和表征的非平凡等价类，或等价类的大小大于1，在$S_n$ $n \geq 7$下的$\{1234, 3412\}$ -等价。这证明了Ma的一个猜想，Ma在研究具有长度为$4$的模式的模式替换等价关系下的$S_n$的非平凡等价类的数量时发现了三个有趣的等价关系，枚举了其中两个关系下的非平凡类，并将上述关于第三个关系下枚举的猜想作为一个开放问题。
英文摘要：
We study the $\{1234, 3412\}$ pattern-replacement equivalence relation on the set $S_n$ of permutations of length $n$, which is conceptually similar to the Knuth relation. In particular, we enumerate and characterize the nontrivial equivalence classes, or equivalence classes with size greater than 1, in $S_n$ for $n \geq 7$ under the $\{1234, 3412\}$-equivalence. This proves a conjecture by Ma, who found three equivalence relations of interest in studying the number of nontrivial equivalence classes of $S_n$ under pattern-replacement equivalence relations with patterns of length $4$, enumerated the nontrivial classes under two of these relations, and left the aforementioned conjecture regarding enumeration under the third as an open problem.
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。
