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Symmetric statistics on rational Dyck paths

Lilan Dai, Shishuo Fu, Dun Qiu

TL;DR

This work extends the study of symmetric joint distributions of path statistics to rational Dyck paths by introducing run and ratio-run alongside ret, and proves composition-type preserving involutions that interchange run with ret (and ratio-run with ret) for $\mathcal{D}_{m,n}$ with $m>n$. A refined rr/\tilde{rr} framework and a hit-and-lift construction underpin these bijections, providing a robust combinatorial mechanism for symmetry. The authors also develop generating-function approaches for both classical and Fuss–Catalan cases, demonstrating explicit symmetry in the associated variables and linking to known combinatorial sequences. Collectively, the results illuminate structural symmetries in rational Dyck-path statistics and suggest broader connections to hyperplane arrangements, parking functions, and Macdonald-polynomial theory, with several natural open problems for further exploration.

Abstract

Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run and ratio-run are introduced. They both have symmetric joint distributions with the return statistic. We give combinatorial proofs and algebraic proofs of the symmetries, generalizing a result of Li and Lin.

Symmetric statistics on rational Dyck paths

TL;DR

This work extends the study of symmetric joint distributions of path statistics to rational Dyck paths by introducing run and ratio-run alongside ret, and proves composition-type preserving involutions that interchange run with ret (and ratio-run with ret) for with . A refined rr/\tilde{rr} framework and a hit-and-lift construction underpin these bijections, providing a robust combinatorial mechanism for symmetry. The authors also develop generating-function approaches for both classical and Fuss–Catalan cases, demonstrating explicit symmetry in the associated variables and linking to known combinatorial sequences. Collectively, the results illuminate structural symmetries in rational Dyck-path statistics and suggest broader connections to hyperplane arrangements, parking functions, and Macdonald-polynomial theory, with several natural open problems for further exploration.

Abstract

Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run and ratio-run are introduced. They both have symmetric joint distributions with the return statistic. We give combinatorial proofs and algebraic proofs of the symmetries, generalizing a result of Li and Lin.

Paper Structure

This paper contains 9 sections, 10 theorems, 43 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized results on rational Dyck paths
  4. The symmetry of (run,ret) distribution on Dmn
  5. The symmetry of (run,ret) distribution on Dmn
  6. Generating function results
  7. Classical Dyck path case
  8. k-Dyck path case
  9. Concluding remarks

Key Result

Theorem 3.1

Given positive integers $m>n$, there is a composition type preserving involution such that $\Phi(\Phi(P))=P$, $\mathrm{run}(\Phi(P))=\mathrm{ret}(P)$ and $\mathrm{ret}(\Phi(P))=\mathrm{run}(P)$.

Figures (5)

  • Figure 1: A $7\times 5$ Dyck path indexed by coarea sequence $(0,0,1,1,3)$.
  • Figure 2: A $15\times 7$ Dyck path.
  • Figure 3: $P\in\mathcal{D}_{15,(1,2,3,2,1,2)}^{(4,2)}$ and $\phi(P)\in\mathcal{D}_{15,(1,2,3,2,1,2)}^{(2,4)}$.
  • Figure 4: $P\in \mathcal{D}_{11, 2111}^{2,\ell}$
  • Figure 5: $P\in\widetilde{\mathcal{D}}_{23,(1,1,2,1,1,1,1,1,1,1),11111110}^{(2,3)}$ and $P'\in\mathcal{D}_{23,(1,1,2,1,1,1,1,1,1,1),11111110}^{(1,4)}$.

Theorems & Definitions (15)

  • Definition : return
  • Definition : run
  • Definition : ratio-run
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • Corollary 3.5
  • proof
  • ...and 5 more