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

TL;DR

This paper enumerates and characterizes the nontrivial equivalence classes of permutations under a specific pattern-replacement relation, confirming a conjecture about their count for permutations of length at least 7.

Contribution

It provides a complete enumeration and characterization of nontrivial classes under the {1234, 3412} pattern-replacement, resolving a conjecture in the field.

Findings

Enumerates nontrivial classes for all n  7.

Characterizes the structure of these classes.

Confirms a conjecture by Ma on class enumeration.

Abstract

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.