Tree-like tableaux

TL;DR

This paper introduces tree-like tableaux, a new combinatorial structure related to permutation and alternative tableaux, providing an insertion procedure that proves they are counted by n! and establishing new bijections with permutations.

Contribution

The paper defines tree-like tableaux, proves they are counted by n!, and introduces two new permutation bijections respecting specific statistics.

Findings

Tree-like tableaux are counted by n! for size n.

An elementary insertion procedure is developed for these tableaux.

Two new bijections between tableaux and permutations are established.

Abstract

In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tree-like tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau.