Associated Permutations of Complete Non-Ambiguous Trees

TL;DR

This paper establishes a bijection between certain combinatorial objects, solves a recurrence relation for their enumeration, and proves a conjecture related to tree-like tableaux, revealing new structural insights and future research directions.

Contribution

It introduces a bijection between tree-like tableaux and a subset of CNATs, and solves a conjecture on their enumeration, advancing combinatorial understanding.

Findings

Established a bijection between tree-like tableaux and CNATs

Solved a recurrence relation for counting tableaux without occupied corners

Proved a conjecture by Laborde-Zubieta

Abstract

We explore new connections between complete non-ambiguous trees (CNATs) and permutations. We give a bijection between tree-like tableaux and a specific subset of CNATs. This map is used to establish and solve a recurrence relation for the number of tree-like tableaux of a fixed size without occupied corners, proving a conjecture by Laborde-Zubieta. We end by establishing a row/column swapping operation on CNATs and identify new areas for future research.