Associated Permutations of Complete Non-Ambiguous Trees and Zubieta's Conjecture

Daniel Chen, Sebastian Ohlig

Published 2022-10-20Version 1

This paper explores connections between complete non-ambiguous trees (CNATs), and permutations. We prove a necessary and sufficient condition for a collection of vertices to be the set of leaves of at least one CNAT, and use this to calculate the number of such collections which are the set of leaves of exactly one CNAT. We give a bijection between tree-like tableaux, in which vertices are placed into a Ferrers diagram, and a certain type of CNAT. This is used to establish and solve a recurrence relation for the number of tree-like tableaux of a given size without occupied corners, proving a conjecture by Zubieta. We end by suggesting an alternative approach for a specific family of CNATs, and identify new areas for research.