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

Abstract

We explore a new type of replacement of patterns in permutations, suggested by James Propp, that does not preserve the length of permutations. In particular, we focus on replacements between 123 and a pattern of two integer elements. We apply these replacements in the classical sense; that is, the elements being replaced need not be adjacent in position or value. Given each replacement, the set of all permutations is partitioned into equivalence classes consisting of permutations reachable from one another through a series of bi-directional replacements. We break the eighteen replacements of interest into four categories by the structure of their classes and fully characterize all of their classes.The terms left-to-right maximum, right-to-left minimum, and right-to-left maximum are defined similarly.We now introduce the classical notion of a pattern.Definition 3. Let π and µ be permutations. A substring p of π forms a copy of the pattern µ if it is order-isomorphic to µ. If such a substring exists, π contains µ. Otherwise, π avoids µ.

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

“…We say that two permutations α, β ∈ S n are P -replacement equivalent if α can be reached from β by a series of P -pattern-replacements. This defines an equivalence relation on S n , which is known as the P -replacement equivalence [9,10,19,12,16,8,15,13,5].…”

“…The study of permutation pattern-replacement equivalences is closely related to the study of permutation pattern avoidance [7,17,4,11,14,1,2,3], which seeks to count the number of singleton equivalence classes (i.e., the number of permutations containing no patterns in P ). The dual problem of counting the number of non-singleton (i.e., nontrivial ) equivalence classes has recently received a large amount of attention in the literature [9,10,19,12,16,8,15,13,5].…”