Bijections from weighted Dyck paths to Schroeder paths

摘要

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