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 subclasses 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 subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.
Highlights
The existence of apparently coincidental equalities 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
We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size n and show that it is exponentially smaller than the nth Catalan number
The Catalan numbers are renowned for their ubiquity in problems of combinatorial enumeration
Summary
The Catalan numbers are renowned for their ubiquity in problems of combinatorial enumeration. We can explain them and many others bijectively – among other things we can show, combining Propositions 13, 14 and 17: The number of 231-avoiding permutations, π, of size n for which the generating function of the class of permutations avoiding both 231 and π is C(n)(t) is the nth Motzkin number. We introduce our basic terminology and notation and consider in more detail a particular quartet of Catalan structures: arch systems, Dyck paths, plane forests, and 231-avoiding permutations. This is followed by some preparatory results before we introduce the relation ∼ and prove its main property, namely that it refines Wilf-equivalence in Theorem 7. We consider some open problems that arise from this work
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