Geodesic
A bijection between the set of nesting-similarity classes and L P matchings
Discrete mathematics & theoretical computer science, Permutation Patterns 2016, Tome 19 (2017-2018) no. 2.

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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. 
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     author = {Martinez, Megan A. and Riehl, Manda},
     title = {A bijection between the set of nesting-similarity classes and {L} & {P} matchings},
     journal = {Discrete mathematics & theoretical computer science},
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     number = {2},
     year = {2017-2018},
     doi = {10.23638/DMTCS-19-2-1},
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Martinez, Megan A.; Riehl, Manda. A bijection between the set of nesting-similarity classes and L & P matchings. Discrete mathematics & theoretical computer science, Permutation Patterns 2016, Tome 19 (2017-2018) no. 2. doi : 10.23638/DMTCS-19-2-1. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-2-1/

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