A Bijection Between Partially Directed Paths in the Symmetric Wedge and Matchings

TL;DR

This paper establishes a bijective correspondence between certain lattice paths and matchings, providing a combinatorial proof for their equinumerosity related to nestings and paths in the symmetric wedge.

Contribution

It introduces a novel bijection linking partially directed paths in a symmetric wedge to matchings, elucidating their combinatorial equivalence.

Findings

Number of paths with k north steps equals number of matchings with k nestings

Provides a bijective proof of a known enumerative result

Connects geometric paths to combinatorial matchings through explicit mapping

Abstract

We give a bijection between partially directed paths in the symmetric wedge y= +/-x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of Prellberg et al. that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y= +/-x with k north steps is equal to the number of matchings on [2n] with k nestings.