  The existence of apparently coincidental equalities (also called
Wilf-equivalences) between the enumeration sequences, or generating functions,
of various hereditary classes of combinatorial structures has attracted
significant interest. We investigate such coincidences among non-crossing
matchings and a variety of other Catalan structures including Dyck paths,
231-avoiding permutations and plane forests. In particular we consider
principal classes defined by not containing an occurrence of a single given
structure. An easily computed equivalence relation among structures is
described such that if two structures are equivalent then the associated
principal classes have the same enumeration sequence. We give an asymptotic
estimate of the number of equivalence classes of this relation among structures
of a given size and show that it is exponentially smaller than the
corresponding Catalan number. In other words these "coincidental" equalities
are in fact very common among principal classes. Our results also allow us to
prove, in a unified and bijective manner, several known Wilf-equivalences from
the literature.
