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Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths

Menghao Qu, Yingrui Zhang

TL;DR

The paper extends the depth statistic to $\vec{k}$-Dyck paths and proves $q,t$-symmetry for the area-depth refined polynomials $\widetilde{C}_{\mathcal{K}}(q,t)$, using a duality on labeled branch trees to construct an area-depth involution. It provides an explicit framework for $\mathcal{K}$-Dyck paths, including three models and the associated statistics, and shows symmetry in broad cases (e.g., last-part independent forms) as well as for the triple $(a,b,c)$ via two independent methods: a direct involution and MacMahon’s partition analysis. The work also clarifies the relationship between $\widetilde{C}_{\vec{k}}(q,t)$ and the standard $C_{\vec{k}}(q,t)$, offering alternative descriptions of higher $q,t$-Catalan polynomials and proposing future directions in polyhedral methods and new statistic pairs. Overall, the results deepen the combinatorial understanding of $q,t$-symmetry in refined Catalan-like polynomials and broaden the toolbox for proving such symmetry beyond the classical Dyck path setting.

Abstract

Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an involution on plane trees. They also provided a new formula for the original $q,t$-Catalan polynomials $C_{n}(q,t)$. We observe that depth is a slight modification of bounce, which was defined by the filling algorithm and ranking algorithm of Xin and the second author in their study of $\vec{k}$-Dyck paths. In this article, we generalize depth of classical Dyck paths to the case of $\vec{k}$-Dyck paths and prove $q,t$-symmetry of the pair of statistics (area, depth) for $\mathcal{K}$-Dyck paths. We provide an alternative description of the higher $q,t$-Catalan polynomials $C_{n}^{(k)}(q,t)$.

Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths

TL;DR

The paper extends the depth statistic to -Dyck paths and proves -symmetry for the area-depth refined polynomials , using a duality on labeled branch trees to construct an area-depth involution. It provides an explicit framework for -Dyck paths, including three models and the associated statistics, and shows symmetry in broad cases (e.g., last-part independent forms) as well as for the triple via two independent methods: a direct involution and MacMahon’s partition analysis. The work also clarifies the relationship between and the standard , offering alternative descriptions of higher -Catalan polynomials and proposing future directions in polyhedral methods and new statistic pairs. Overall, the results deepen the combinatorial understanding of -symmetry in refined Catalan-like polynomials and broaden the toolbox for proving such symmetry beyond the classical Dyck path setting.

Abstract

Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are -symmetric by constructing an involution on plane trees. They also provided a new formula for the original -Catalan polynomials . We observe that depth is a slight modification of bounce, which was defined by the filling algorithm and ranking algorithm of Xin and the second author in their study of -Dyck paths. In this article, we generalize depth of classical Dyck paths to the case of -Dyck paths and prove -symmetry of the pair of statistics (area, depth) for -Dyck paths. We provide an alternative description of the higher -Catalan polynomials .

Paper Structure

This paper contains 17 sections, 21 theorems, 59 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background and Definitions
  3. Three models of $\vec{k}$-Dyck paths
  4. Combinatorial statistics of $\vec{k}$-Dyck paths
  5. Refined $q,t$-Catalan polynomials of $\vec{k}$-Dyck paths
  6. Area-Bounce Polynomials
  7. Area-Depth polynomials
  8. $q,t$-symmetry of $\widetilde{C}_{\mathcal{K}}(q,t)$
  9. Labeled branch trees
  10. Proof of Theorem \ref{['thm-aK']}.
  11. $q,t$-symmetry of $\widetilde{C}_{\vec{k}}(q,t)$
  12. $q,t$-symmetry of $\widetilde{C}_{\vec{k}}(q,t)$ for some special cases.
  13. Relationship between $C_{\vec{k}}(q,t)$ and $\widetilde{C}_{\vec{k}}(q,t)$.
  14. $q,t$-symmetry of $\widetilde{C}_{\vec{k}}(q,t)$ for $\vec{k}=(a,b,c)$
  15. Proof via an explicit involution
  16. ...and 2 more sections

Key Result

Theorem 1.1

$\widetilde{C}_{^a\mathcal{K}}(q,t)$ is $q,t$-symmetric for any $\vec{k}$ and positive $a$.

Figures (8)

  • Figure 1: $\vec{k}=(4,2,3,1)$ and $\pi=S^4 W S^2 W W W W S^3 W W S^1 W W W$.
  • Figure 2: Filling tableau $\eta(\pi)$ and Ranking tableau $\gamma(\eta(\pi))$.
  • Figure 3: Depth labeling of $\pi=NNEENNNEENEENNEE$.
  • Figure 4: Filling tableau $\eta_{*}(\pi)$ and Ranking tableau $\gamma_{*}(\eta_{*}(\pi))$.
  • Figure 5: Filling tableau $\eta_{*}(\pi)$ and Ranking tableau $\gamma_{*}(\eta_{*}(\pi))$.
  • ...and 3 more figures

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Example 1.5
  • Definition 2.1: Classical path model
  • Definition 2.2: Visual path model
  • Definition 2.3: Word model
  • Example 2.4
  • Example 2.5
  • ...and 64 more