Partitions and Partial Matchings Avoiding Neighbor Patterns

TL;DR

This paper derives generating functions for specific partial matchings avoiding certain neighbor patterns and establishes bijections linking these matchings to set partitions and permutation classes, revealing deep combinatorial connections.

Contribution

It introduces new generating functions for neighbor pattern-avoiding partial matchings and constructs bijections connecting these matchings to well-studied combinatorial objects.

Findings

Generated explicit formulas for neighbor pattern-avoiding partial matchings.

Established bijections between matchings, set partitions, and permutations.

Connected neighbor pattern avoidance to known combinatorial structures.

Abstract

We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding three neighbor patterns (neighbor alignments, left nestings and right nestings) and set partitions avoiding right nestings via an intermediate structure of integer compositions. Such integer compositions are known to be in one-to-one correspondence with self-modified ascent sequences or $3\bar{1}52\bar{4}$-avoiding permutations, as shown by Bousquet-M\'elou, Claesson, Dukes and Kitaev.