New Results on Doubly Adjacent Pattern-Replacement Equivalences

TL;DR

This paper advances the understanding of pattern-replacement equivalences in permutations, solving five open problems by deriving formulas and representatives for classes, and generalizing results to broader partitions and equivalence relations.

Contribution

It provides formulas and representatives for several unresolved pattern-replacement equivalences, generalizes results to all partitions of S_3, and introduces a confluence method for class characterization.

Findings

Formulas for three of the five key equivalences.

Systems of representatives for the remaining two equivalences.

A generalized characterization of equivalence classes in S_n.

Abstract

In this paper, we consider the family of pattern-replacement equivalence relations referred to as the "indices and values adjacent" case. Each such equivalence is determined by a partition $P$ of a subset of $S_c$ for some $c$. In 2010, Linton, Propp, Roby, and West posed a number of open problems in the area of pattern-replacement equivalences. Five, in particular, have remained unsolved until now, the enumeration of equivalence classes under the $\{123, 132\}$-equivalence, under the $\{123, 321\}$-equivalence, under the $\{123, 132, 213\}$ equivalence, and under the $\{123, 132, 213, 321\}$-equivalence. We find formulas for three of the five equivalences and systems of representatives for the equivalence classes of the other two. We generalize our results to hold for all replacement partitions of $S_3$, as well as for an infinite family of other replacement partitions. In addition, we characterize the equivalence classes in $S_n$ under the $S_c$-equivalence, finding a generalization of Stanley's results on the $\{12, 21\}$-equivalence. To do this, we introduce a notion of confluence that often allows one to find a representative element in each equivalence class under a given equivalence relation. Using an inclusion-exclusion argument, we are able to use this to count the equivalence classes under equivalence relations satisfying certain conditions.