A general theory of Wilf-equivalence for Catalan structures
  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. 
Michael Albert and Mathilde Bouvel
math.CO