论文标题
计算$ \ {1234,3412 \} $ $ s_n $的非平凡等效类
Counting the Nontrivial Equivalence Classes of $S_n$ under $\{1234,3412\}$-Pattern-Replacement
论文作者
论文摘要
我们研究了$ \ {1234，3412 \} $模式替换等价关系，在长度$ n $的集合$ s_n $上，这与Knuth的关系在概念上相似。特别是，我们列举并表征非平凡的等价类，或大小大于1的等价类别，$ s_n $ in $ n \ geq 7 $下的$ \ {1234，3412 \} $ - 等价。这证明了MA的猜想，他在研究模式替换等效关系的情况下与$ s_n $的非平价等效类别的数量与长度$ 4 $的模式进行了$ 4 $的$ s_n $的数量，在这些关系的两个关系下列举了非平凡的类别，并使对第三次宣布第三次估计不足的猜测。
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.
