Tree-like tableaux

Abstract

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.

Stammering tableaux

“…In particular, there is the question of a recursive construction to make clear that there are n! such objects of size n, which can involve insertion algorithms [2,3]. As we will see through this article, the recursive construction in our case is extremely simple.…”

“…Later, other kinds of tableaux have been introduced: alternative tableaux [22], tree-like tableaux [2], and Dyck tableaux [3]. They are all variants of each other, but each has its own combinatorial properties.…”

“…Tree-like tableaux [2] have a nice tree structure, and there is a nice insertion algorithm that permits building these object inductively. Particularly relevant to the present work, Dyck tableaux [3] connects with labelled Dyck paths called Laguerre histories (see also [11]). These various refecences make clear that all this tableau combinatorics is interesting in itself, and the link with the PASEP is sometimes not apparent.…”

Non-ambiguous trees: New results and generalisation

“…Tree-like tableaux [ABN13] are certain fillings of Ferrers diagram, in simple bijection with permutations or alternative tableaux [Pos07,Vie08]. They are the subject of an intense research activity in combinatorics, mainly because they appear as the key tools in the combinatorial interpretation of the well-studied model of statistical mechanics called PASEP: they naturally encode the states of the PASEP, together with the transition probabilities through simple statistics [CW07].…”

“…In the general case, the probability of a given state of the model depends on five parameters (q, α, β, γ, δ) which give the probability of transitions (a particle moving to the right or the left, or going in or out the model, when possible). Tree-like tableaux [ABN13] have been proven to give a combinatorial interpretation for the steady state of the PASEP when γ = δ = 0, the weight in α (resp. β) corresponding to the statistic L 0 (B) (resp.…”