Counting the Nontrivial Equivalence Classes of $S_n$ under $\{1234,3412\}$-Pattern-Replacement

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.