Nestings of Matchings and Permutations and North Steps in PDSAWs

TL;DR

This paper establishes bijections linking matchings and permutations with nestings to specific lattice paths, revealing their equinumerous nature and enabling parameter transfer.

Contribution

It provides simple bijective proofs connecting matchings and permutations with PDSAWs, highlighting their equinumerous counts and parameter correspondences.

Findings

Matchings with N nestings are equinumerous to PDSAWs with N north steps.

Permutations with N nestings correspond to PDSAWs below the x-axis.

Bijections preserve several combinatorial parameters.

Abstract

We present a simple bijective proof of the fact that matchings of [2n] with N nestings are equinumerous to partially directed self avoiding walks confined to the symmetric wedge defined by y=+-x, with n east steps and N north steps. A very similar construction connects permutations with N nestings and PDSAWs remaining below the x-axis, again with N north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.