Pattern avoidance in the matching pattern poset

TL;DR

This paper studies the pattern avoidance in matchings, providing explicit formulas and recursive methods for enumerating classes of matchings that avoid certain patterns, including new and lifted patterns, and introduces unlabeled patterns with combinatorial enumeration.

Contribution

It extends previous work by deriving explicit formulas and recursive generating functions for pattern-avoiding matchings, including new pattern classes and unlabeled patterns, with combinatorial interpretations.

Findings

Explicit formulas for matchings avoiding two new patterns.

Recursive formula for matchings avoiding lifted patterns.

Enumeration of matchings avoiding unlabeled patterns using bijections.

Abstract

A matching of the set $[2n]=\{ 1,2,\ldots ,2n\}$ is a partition of $[2n]$ into blocks with two elements, i.e. a graph on $[2n]$ such that every vertex has degree one. Given two matchings $\sigma$ and $\tau$ , we say that $\sigma$ is a pattern of $\tau$ when $\sigma$ can be obtained from $\tau$ by deleting some of its edges and consistently relabelling the remaining vertices. This is a partial order relation turning the set of all matchings into a poset, which will be called the matching pattern poset. In this paper, we continue the study of classes of pattern avoiding matchings, initiated by Chen, Deng, Du, Stanley and Yan (2007), Jelinek and Mansour (2010), Bloom and Elizalde (2012). In particular, we work out explicit formulas to enumerate the class of matchings avoiding two new patterns, obtained by juxtaposition of smaller patterns, and we describe a recursive formula for the generating function of the class of matchings avoiding the lifting of a pattern and two additional patterns. Finally, we introduce the notion of unlabeled pattern, as a combinatorial way to collect patterns, and we provide enumerative formulas for two classes of matchings avoiding an unlabeled pattern of order three. In one case, the enumeration follows from an interesting bijection between the matchings of the class and ternary trees.