A general theory of Wilf-equivalence for Catalan structures

TL;DR

This paper develops a unified theory explaining why different Catalan-structured classes often have identical enumeration sequences, revealing that such Wilf-equivalences are surprisingly common and systematically characterized.

Contribution

It introduces an equivalence relation among Catalan structures that predicts Wilf-equivalences and provides asymptotic estimates of their prevalence among classes of a given size.

Findings

The equivalence relation classifies structures with identical enumeration sequences.

The number of equivalence classes grows exponentially slower than Catalan numbers.

The framework unifies and explains several known Wilf-equivalences.

Abstract

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.