  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.
