  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.
