Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections

TL;DR

This paper explores duality-preserving bijections among non-crossing trees, quadrangular dissections, and ternary trees using the theory of Properly Embedded Graphs, revealing new combinatorial correspondences and formulas for counting symmetric structures.

Contribution

It introduces a structural bijection between quadrangular dissections and non-crossing trees, and generalizes duality in free ternary magmas with explicit enumeration formulas.

Findings

Defined an involutory duality on labeled non-crossing trees

Established a new bijection between quadrangular dissections and non-crossing trees

Derived formulas for counting self-dual structures

Abstract

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the set of ternary trees, the set of quadrangular dissections, and the set of flagged Perfectly Chain Decomposed Ditrees, and we give topological and/or combinatorial interpretations of the structural bijections between them. In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new. Further we give explicit formulas for the number of self-dual labeled and unlabeled non-crossing trees and the set of quadrangular dissections up to rotations and up to rotations and reflections.