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.

New Results on Pattern-Replacement Equivalences: Generalizing a Classical Theorem and Revising a Recent Conjecture

“…Our second main result extends the work of [6] on rotational equivalences. Kuszmaul and Zhou proved that the rotational equivalences always yield either one or two nontrivial equivalence classes in S n , and conjectured that the number of nontrivial classes depended only on the patterns involved in the rotational equivalence (rather than on n) [8]. We present a counterexample to their conjecture, and prove a new theorem fully classifying (for large n) when there is one nontrivial equivalence class and when there are two nontrivial equivalence classes.…”

“…In this paper, we study what are known as pattern-replacement equivalences [2,[6][7][8]13,14]. Consider a set of patterns such as {123, 231}.…”

“…Subsequent research [6,7] continued to focus on the case of patterns of length 3. Then, in 2013, [8] presented the first results on infinite families of permutation patterns. They showed that for any set Π consisting of some pattern and its cyclic shifts, the number of nontrivial equivalence classes in S n under Π-equivalence with adjacency constraints is at most two.…”

Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns

“…However, this misses id(4). We now show id(4) ≡ id (7) The replacements 123 ↔ * 23, 123 ↔ 1 * 3, and 123 ↔ 12 * simply drop or add an element in a 123 pattern. In the remainder of this section we will proceed simultaneously with 12 * and 1 * 3 by using γ to denote an arbitrary selection from the two, and later use the reverse complement symmetry to state the result of equivalence classes for 123 ↔ * 23.…”

“…Together these three papers enumerate the equivalence classes in a general S n and count the size of the class containing the identity for almost all cases. In addition, William Kuszmaul and Ziling Zhou examine in [7] equivalence classes under more general families of replacements. Throughout all of this work, permutation length was preserved under replacements.…”