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Promotion and rowmotion in rational Catalan combinatorics

Keiichi Shigechi

Abstract

We study four bijections, which are promotion, evacuation, rowmotion, and rowvacuation, on generalized Dyck paths in rational Catalan combinatorics. We define the maps on generalized Dyck paths, which have their origins in maps on Dyck paths and non-crossing partitions. They include rotation, Kreweras complement map, Simion--Ullman involution on non-crossing partitions, and Lalanne--Kreweras involution on Dyck paths. These maps have an expression in terms of the four combinatorial bijections. By extending the bijection studied by D. Armstrong, C. Stump, and H. Thomas on one hand, and the correspondence of RSK type studied by B. Adenbaum and S. Elizalde on the other, we present the equivalence between the two bijections, promotion and rowmotion, on generalized Dyck paths through these bijection and correspondence. For this purpose, we provide an alternative description of the correspondence of RSK type in terms of Dyck tilings.

Promotion and rowmotion in rational Catalan combinatorics

Abstract

We study four bijections, which are promotion, evacuation, rowmotion, and rowvacuation, on generalized Dyck paths in rational Catalan combinatorics. We define the maps on generalized Dyck paths, which have their origins in maps on Dyck paths and non-crossing partitions. They include rotation, Kreweras complement map, Simion--Ullman involution on non-crossing partitions, and Lalanne--Kreweras involution on Dyck paths. These maps have an expression in terms of the four combinatorial bijections. By extending the bijection studied by D. Armstrong, C. Stump, and H. Thomas on one hand, and the correspondence of RSK type studied by B. Adenbaum and S. Elizalde on the other, we present the equivalence between the two bijections, promotion and rowmotion, on generalized Dyck paths through these bijection and correspondence. For this purpose, we provide an alternative description of the correspondence of RSK type in terms of Dyck tilings.
Paper Structure (36 sections, 48 theorems, 150 equations, 8 figures)

This paper contains 36 sections, 48 theorems, 150 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Dyck paths and rational Dyck paths
  4. Maps on a Dyck path
  5. RSK and 321-avoiding permutations
  6. Promotion and evacuation
  7. General theory
  8. Promotion and evacuation on Dyck paths
  9. Promotion and evacuation on rational Dyck paths
  10. Rowmotion and rowvacuation
  11. General theory
  12. Rowmotion and Rowvacuation on Dyck paths
  13. Matching map and RSK correspondence
  14. Matching map
  15. Crossing perfect matchings and RSK correspondence
  16. ...and 21 more sections

Key Result

Proposition 2.6

We have $|\mathtt{SYT}_{(a,b)}(n)|=|\mathtt{Dyck}_{(a,b)}(n)|$.

Figures (8)

  • Figure 2.3: Examples of $2$-Dyck paths
  • Figure 2.9: A perfect matching of a $(2,3)$-Dyck path. The rational Dyck path on the left corresponds to the perfect matching $\{\{1,4,10\},\{2,3\},\{5,6,9\},\{7,8\}\}$.
  • Figure 6.3: The non-crossing partition $\{\{1,3\},\{2\},\{4,5,7\},\{6\}\}$
  • Figure 6.6: An example of weighted non-crossing partition of $[4]$.
  • Figure 2.12: The map $\mathtt{dPM}$ on a $(2,3)$-Dyck path. The rational Dyck path on the left corresponds to the perfect matching $\{\{1,10\},\{2,4\},\{3\},\{5,7,9\},\{6\},\{8\}\}$
  • ...and 3 more figures

Theorems & Definitions (138)

  • Definition 2.1
  • Definition 2.2
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.10
  • Definition 2.11
  • Definition 2.13
  • ...and 128 more