Authors: Jean-Christophe Aval ¹; Adrien Boussicault ¹; Philippe Nadeau ²

• 1 Laboratoire Bordelais de Recherche en Informatique
• 2 Faculty of Mathematics [Vienna]

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 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.