Tree-like tableaux

Jean-Christophe Aval (1) , Adrien Boussicault (1) , Philippe Nadeau (2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 ICJ - Institut Camille Jordan

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.

Mots clés: Tree-like tableaux; Permutation tableaux; Alternative tableaux; PASEP

Domaines: Combinatoire [math.CO]

Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau. Tree-like tableaux. The Electronic Journal of Combinatorics, 2013, 20, pp.P34. ⟨hal-00618274v2⟩