About the Determinant of Complete non-ambiguous Trees

Complete non-ambiguous trees (CNATs) are combinatorial objects which appear in various contexts.Recently, Chen and Ohlig studied the notion of leaf permutation on these objects, and proposed a series of nice conjectures.Most of them were proved by Selig and Zhu, through a connexion with the abelian sandpile model.But one conjecture remained open, about the distribution of a natural statistic named determinant.We prove this conjecture, in a bijective way.