Permutation-generated maps between Dyck paths

TL;DR

This paper generalizes permutation-induced bijections on Dyck paths, characterizing permutations that produce identical maps and providing new combinatorial interpretations and statistics related to Dyck paths.

Contribution

It extends previous work by allowing arbitrary permutations to generate Dyck path maps and characterizes when different permutations produce the same bijection.

Findings

Identifies permutations generating the same Dyck path map

Provides a new combinatorial interpretation of (2n-1)!!

Introduces new Dyck path statistics with known height distributions

Abstract

In 2003, Deutsch and Elizalde defined a family of bijective maps between the set of Dyck paths to itself which is induced by some particular permutations. In this paper, we extend the construction of the maps by allowing the permutation to be arbitrary. We characterise the permutations which generate the same map and find all permutations generating a bijection among Dyck paths. Consequently, we give a new combinatorial interpretation of the quantity $(2n-1)!!$ as well as some new statistics of Dyck paths which are equidistributed to some known height statistics via our generalised maps.