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

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.Lemma 2.17. Every permutation not ending with 1 is equivalent to a permutation of the form . . . 1j for some j under the {123, 132, 312}-equivalence.Proof. Such a permutation is reached through repeated applications of 123 → 312 and 132 → 312 using the actual letter 1.

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

“…We may immediately characterize the classes with the following theorem. In the proof, we draw inspiration from the stooge sort, in a manner similar to the proof of Proposition 2.17 in [6].…”

“…They inspect these replacements when applied to elements in general position, elements with adjacent positions, and elements with adjacent positions and adjacent values. A couple papers, [11], by James Propp et al, and [6], by William Kuszmaul, follow up on replacements on elements with adjacent positions. 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.…”

New results on families of pattern-replacement equivalences

“…Let T c be the set {|T c, j | : j > 0}. 1 We follow the convention that id n = 123 · · · n. It follows from Theorem 4.9 and Theorem 4.5 that the size of any equivalence class under the S c -equivalence is a product of elements of T c . Consequently, it is interesting to study |T c,n |.…”

“…Let T c be the set {|T c, j | : j > 0}. 1 We follow the convention that id n = 123 • • • n. 3 . Now consider the permutations in S n comprising only of irreducible blocks of size ≤ 3 but containing no consecutive irreducible blocks whose sizes add to 3.…”

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

“…Both the Knuth and the forgotten relation have a common structure: They are transitive, reflexive, symmetric closures of relations given by allowing the rearrangement of blocks of adjacent letters in a permutation from a certain order into another given order. This inspired various authors [3], [5], [2] to systematically analyze relations of the same type, leading to numerous results on the number of equivalence classes and their sizes. This paper extends the study to consider pattern-replacement relations using patterns of arbitrary length.…”