Bijections from weighted Dyck paths to Schroeder paths

TL;DR

This paper introduces bijections between weighted Dyck paths and Schroeder paths, providing new insights into their combinatorial structures and connections to permutation classes.

Contribution

It presents explicit bijections from Dyck paths to both little and big Schroeder paths, expanding understanding of their combinatorial relationships.

Findings

Bijections between Dyck paths and Schroeder paths are constructed.

Generating functions for new classes of lattice paths are derived.

A relationship with 231-avoiding permutations is established.

Abstract

Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schroeder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schroeder paths, and from a similar set of Dyck paths to big Schroeder paths. We show the effect of these bijections on the corresponding matchings, find generating functions for two new classes of lattice paths, and demonstrate a relationship with 231-avoiding permutations.