Combinatorics of non-ambiguous trees

TL;DR

This paper explores the combinatorial properties of non-ambiguous trees, providing enumeration formulas, combinatorial proofs of known identities, a hook formula, and a bijection with parallelogram polyominoes.

Contribution

It introduces new combinatorial insights into non-ambiguous trees, including enumeration, identities, a hook formula, and a bijection with polyominoes.

Findings

Enumeration formulas for non-ambiguous trees

Combinatorial proofs of identities by Carlitz and Ehrenborg-Steingrímsson

A bijection between parallelogram polyominoes and binary trees

Abstract

This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingr\'imsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.