Non-ambiguous trees: New results and generalisation

Abstract

We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yields a new formula for the number of NATs. We also obtain q-versions of our formula. We finally generalise NATs to higher dimension.

About the Determinant of Complete non-ambiguous Trees

“…They were put to light as a special case of tree-like tableaux, which have been found to have applications in the PASEP model of statistical mechanics [5,3]. The initial study of NATs revealed nice properties, mostly in an enumerative context [1,2]. This includes enumeration formulas with respect to fixed constraints (hook formula), and new bijective proofs of combinatorial identities.…”