Equivalence Classes in $S_n$ for Three Families of Pattern-Replacement Relations

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 $S_c$. In particular, we are interested in the number of classes created in $S_n$ by each relation and in characterizing these classes. Imposing the condition that the partition of $S_c$ has one nontrivial part containing the cyclic shifts of a single permutation, we find enumerations for the number of nontrivial classes. When the permutation is the identity, we are able to compare the sizes of these classes and connect parts of the problem to Young tableaux and Catalan lattice paths. Imposing the condition that the partition has one nontrivial part containing all of the permutations in $S_c$ beginning with 1, we both enumerate and characterize the classes in $S_n$. We do the same for the partition that has two nontrivial parts, one containing all of the permutations in $S_c$ beginning with 1, and one containing all of the permutations in $S_c$ ending with 1.

References in corpus (2)

• The forgotten monoid
• Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions

Cited by in corpus (4)

• Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns
• New Results on Doubly Adjacent Pattern-Replacement Equivalences
• New Results on Pattern-Replacement Equivalences: Generalizing a Classical Theorem and Revising a Recent Conjecture
• A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions