Dinv and Area

TL;DR

This paper provides a new combinatorial proof of the identity linking dinv and area for (m,n)-Dyck paths, expanding understanding of these combinatorial objects and their properties.

Contribution

It introduces a novel combinatorial proof of the dinv-area equality for (m,n)-Dyck paths, complementing existing proofs.

Findings

New combinatorial proof of dinv equals area for (m,n)-Dyck paths

Extends understanding of Dyck path statistics

Supports existing identities with an alternative approach

Abstract

We give a new combinatorial proof of the well known result that the dinv of an $(m,n)$-Dyck path is equal to the area of its sweep map image. The first proof of this remarkable identity for co-prime $(m,n)$ is due to Loehr and Warrington. There is also a second proof (in the co-prime case) due to Gorsky and Mazin and a third proof due to Mazin.