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Tamari intervals and blossoming trees

Wenjie Fang, Éric Fusy, Philippe Nadeau

TL;DR

The paper introduces a direct bijection Phi between Tamari intervals and blossoming trees via a meandering-diagram representation, uniting interval- and map-encodings. It shows Phi commutes with duality and specializes cleanly to synchronized, Kreweras, modern/new, infinitely modern, and self-dual subfamilies, yielding concise combinatorial proofs of their counts. The framework recovers known formulas, derives new ones for self-dual cases, and links to interval-posets, cubic coordinates, and the Poulalhon–Schaeffer closure. A dual involution rho is presented, and practical implementation aids are provided for experimentation. The work advances understanding of Tamari interval structures, duality interplay, and connections to planar maps and Catalan-combinatorics families.

Abstract

We introduce a simple bijection between Tamari intervals and the blossoming trees (Poulalhon and Schaeffer, 2006) encoding planar triangulations, using a new meandering representation of such trees. Its specializations to the families of synchronized, Kreweras, new/modern, and infinitely modern intervals give a combinatorial proof of the counting formula for each family. Compared to (Bernardi and Bonichon, 2009), our bijection behaves well with the duality of Tamari intervals, enabling also the counting of self-dual intervals.

Tamari intervals and blossoming trees

TL;DR

The paper introduces a direct bijection Phi between Tamari intervals and blossoming trees via a meandering-diagram representation, uniting interval- and map-encodings. It shows Phi commutes with duality and specializes cleanly to synchronized, Kreweras, modern/new, infinitely modern, and self-dual subfamilies, yielding concise combinatorial proofs of their counts. The framework recovers known formulas, derives new ones for self-dual cases, and links to interval-posets, cubic coordinates, and the Poulalhon–Schaeffer closure. A dual involution rho is presented, and practical implementation aids are provided for experimentation. The work advances understanding of Tamari interval structures, duality interplay, and connections to planar maps and Catalan-combinatorics families.

Abstract

We introduce a simple bijection between Tamari intervals and the blossoming trees (Poulalhon and Schaeffer, 2006) encoding planar triangulations, using a new meandering representation of such trees. Its specializations to the families of synchronized, Kreweras, new/modern, and infinitely modern intervals give a combinatorial proof of the counting formula for each family. Compared to (Bernardi and Bonichon, 2009), our bijection behaves well with the duality of Tamari intervals, enabling also the counting of self-dual intervals.
Paper Structure (33 sections, 31 theorems, 30 equations, 27 figures, 1 table)

This paper contains 33 sections, 31 theorems, 30 equations, 27 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tamari intervals and their meandering representation
  3. Tamari lattice and intervals
  4. Some representations and encodings of binary trees
  5. From pairs of binary trees to meandering diagrams/trees
  6. Proof of Proposition \ref{['prop:phi']}
  7. Connection with previous work
  8. Relation with interval-posets
  9. Relation to cubic coordinates for Tamari intervals
  10. Blossoming trees and their meandering representation
  11. The main bijection, properties and specializations
  12. Parameter-correspondence
  13. Commutation with duality of intervals
  14. Specializations
  15. Synchronized intervals
  16. ...and 18 more sections

Key Result

Theorem 1.1

The bijection $\Phi$ between intervals in $\mathrm{Tam}_n$ and bicolored blossoming trees of size $n$ sends self-dual intervals to blossoming trees with a half-turn symmetry. Its specializations to synchronized, Kreweras, modern/new, and infinitely modern intervals yield combinatorial proofs of coun

Figures (27)

  • Figure 1: A binary tree $T$ with its canonical drawing and smooth drawing.
  • Figure 2: Construction of the diagram-drawing (with the degree-vector indicated below) from the smooth drawing.
  • Figure 3: A binary tree $T$, its bracket-vector $\mathbf{V}(T)$ and dual bracket-vector $\mathbf{V}'(T)$.
  • Figure 4: A pair $(T,T')$ of binary trees of same size, its canonical drawing, and its smooth drawing.
  • Figure 5: The operation of $\phi$ on the smooth drawing of a pair $(T,T') \in \mathcal{X}_n$, performed at each segment between consecutive black points on the horizontal axis. The smaller blue arc and red arc in the left figure may be reduced to a point.
  • ...and 22 more figures

Theorems & Definitions (83)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • ...and 73 more