# A bijection between the set of nesting-similarity classes and L & P matchings

## TL;DR

This paper establishes a bijection between nesting-similarity classes and L & P matchings, linking RNA secondary structure models with combinatorial classes, and preserving key structural properties.

## Contribution

It introduces a bijection between nesting-similarity classes and L & P matchings, connecting two enumerated combinatorial structures relevant to RNA modeling.

## Key findings

- Bijection preserves noncrossing matchings
- Bijection maintains sequence of edge beginnings and endings
- Links two classes of matchings with equal enumeration

## Abstract

Matchings are frequently used to model RNA secondary structures; however, not all matchings can be realized as RNA motifs. One class of matchings, called the L $\&$ P matchings, is the most restrictive model for RNA secondary structures in the Largest Hairpin Family (LHF). The L $\&$ P matchings were enumerated in $2015$ by Jefferson, and they are equinumerous with the set of nesting-similarity classes of matchings, enumerated by Klazar. We provide a bijection between these two sets. This bijection preserves noncrossing matchings, and preserves the sequence obtained reading left to right of whether an edge begins or ends at that vertex.