Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

TL;DR

This paper investigates permutation equivalence classes under pattern-replacement relations inspired by Knuth and forgotten relations, providing formulas and enumerations for various partitions of three-letter permutation groups.

Contribution

It introduces new formulas and enumerations for permutation classes under pattern-replacement relations based on partitions of $S_3$, extending previous work on Knuth and forgotten relations.

Findings

Formulas for the number of classes in previously unsolved cases.

Enumeration of classes for 13 of 14 unresolved cases.

Some enumerations match those of Knuth and forgotten relations, unexplained phenomena.

Abstract

We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previously unsolved case. When the partition is of $S_3$ and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for $13$ of the $14$ unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery.