International audience 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. Dans ce travail nous introduisons et étudions les tableaux boisés, qui sont certains remplissages de diagrammes de Ferrers en bijection simple avec les tableaux de permutation et les tableaux alternatifs. Nous décrivons une procédure d'insertion élémentaire sur nos tableaux qui donne une preuve limpide que les tableaux de taille n sont comptés par n!, et qui de plus respecte la plupart des statistiques standard sur les tableaux de permutation et tableaux alternatifs. Notre procédure d'insertion permet en particulier de définir deux nouvelles bijections simples entre tableaux et permutations: la première est conçue spécifiquement pour respecter le motif généralisé 2-31 sur les permutations, tandis que la deuxième respecte l'arbre binaire sous-jacent à un tableau boisé.

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.All authors are supported by the ANR (PSYCO project -ANR-11-JS02-001).

International audienceThis 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 Steingrímsson. 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