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On the area-depth symmetry on Łukasiewicz paths

Wenjie Fang

TL;DR

This work settles a conjecture on $q,t$-symmetry for area and depth statistics in a broad class of Łukasiewicz-path generalizations of Dyck paths by leveraging the classical plane-tree bijection. By translating $\mathtt{area}$ and $\mathtt{depth}$ to tree statistics $\mathtt{rthorn}$ and $\mathtt{lthorn}$, and introducing lodestar-based involutions, the authors provide transparent combinatorial proofs of symmetry for $\widetilde{C}_{a,M}(q,t)$ and $\widetilde{C}_{M,b}(q,t)$, and extend these results to $\widetilde{C}_{a,M,b}(q,t)$. The approach yields a direct, structurally motivated explanation of $q,t$-symmetry, along with corollaries and a generating-function formulation that generalizes existing $q,t$-Catalan frameworks. Overall, the paper deepens understanding of how area and depth interact under tree-based representations, with implications for $q,t$-symmetric polynomials in related path families.

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 Łukasiewicz paths by interpreting $\mathtt{area}$ and $\mathtt{depth}$ under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent.

On the area-depth symmetry on Łukasiewicz paths

TL;DR

This work settles a conjecture on -symmetry for area and depth statistics in a broad class of Łukasiewicz-path generalizations of Dyck paths by leveraging the classical plane-tree bijection. By translating and to tree statistics and , and introducing lodestar-based involutions, the authors provide transparent combinatorial proofs of symmetry for and , and extend these results to . The approach yields a direct, structurally motivated explanation of -symmetry, along with corollaries and a generating-function formulation that generalizes existing -Catalan frameworks. Overall, the paper deepens understanding of how area and depth interact under tree-based representations, with implications for -symmetric polynomials in related path families.

Abstract

In an effort to further understanding -Catalan statistics, a new statistic on Dyck paths called was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known statistics. In a recent preprint, Qu and Zhang (2025) generalized to so-called ``-Dyck paths''. They showed that and 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 Łukasiewicz paths by interpreting and under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent.
Paper Structure (4 sections, 8 theorems, 5 equations, 4 figures)

This paper contains 4 sections, 8 theorems, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Łukasiewicz paths and its statistics
  3. Plane trees and $q,t$-statistics
  4. Symmetries in $q,t$ through trees

Key Result

Theorem 1.1

We define $\widetilde{C}_{a,M,b}(q, t)$ as a sum over Łukasiewicz paths $P$ with the first (resp. last) up-step of degree $a$ (resp. $b$) and $M$ the multiset of the degrees of other up-steps: Then $\widetilde{C}_{a,M,b}(q, t)$ is symmetric in $q, t$, i.e., $\widetilde{C}_{a, M, b}(q, t) = \widetilde{C}_{a, M, b}(t, q)$.

Figures (4)

  • Figure 1: Example of a Łukasiewicz path, with its statistics $\mathtt{area}$ and $\mathtt{depth}$.
  • Figure 2: Example of the bijections $\lambda$ and $\tau$.
  • Figure 3: (a) Illustration of $\mathtt{lthorn}$ and $\mathtt{rthorn}$ on an internal node. (b) Example of the $\mathtt{lthorn}$. (c) Example of $\mathtt{rthorn}$.
  • Figure 4: Example of the lodestar swapping map.

Theorems & Definitions (21)

  • Theorem 1.1: kvec-depth
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • ...and 11 more