Restricting Dyck Paths and 312-avoiding Permutations

TL;DR

This paper explores a combinatorial interpretation linking restricted Dyck paths and 312-avoiding permutations, providing recursive enumeration formulas and identities involving Catalan numbers.

Contribution

It introduces a new interpretation of restricted Dyck paths via 312-avoiding permutations with specific maxima restrictions and develops recursive enumeration methods.

Findings

Established a bijection between restricted Dyck paths and 312-avoiding permutations.

Derived recursive formulas for counting these structures.

Produced combinatorial identities involving Catalan numbers.

Abstract

Dyck paths having height at most $h$ and without valleys at height $h-1$ are combinatorially interpreted by means of 312-avoding permutations with some restrictions on their \emph{left-to-right maxima}. The results are obtained by analyzing a restriction of a well-known bijection between the sets of Dyck paths and 312-avoding permutations. We also provide a recursive formula enumerating these two structures using ECO method and the theory of production matrices. As a further result we obtain a family of combinatorial identities involving Catalan numbers.