On the area-depth symmetry on {\L}ukasiewicz paths

TL;DR

This paper proves a symmetry property between two statistics, area and depth, on Lukasiewicz paths by using a bijection to plane trees, confirming a conjecture in a more general setting.

Contribution

It establishes the area-depth symmetry for Lukasiewicz paths, extending previous results and confirming a conjecture in a broader combinatorial context.

Findings

Proves the area-depth symmetry on Lukasiewicz paths.

Uses bijection to plane trees to demonstrate symmetry.

Confirms a conjecture about { extbackslash}vec{k-}Dyck paths in a general setting.

Abstract

In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\mathtt{depth}$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $\mathtt{area}$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $\mathtt{depth}$ to so-called ``$\vec{k}$-Dyck paths''. They showed that $\mathtt{area}$ and $\mathtt{depth}$ are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of {\L}ukasiewicz paths by interpreting $\mathtt{area}$ and $\mathtt{depth}$ under the classical bijection between {\L}ukasiewicz paths and plane trees, through which the symmetry is transparent.