Dyck tableaux

The starting point of this work is the discovery of a new and direct construction that relies bijectively the permutations of length n to some weighted Dyck paths named subdivided Laguerre histories. These objects correspond to the combinatorial interpretation of the development of the generating function for factorial numbers in terms of a Stieltjes continued fraction [9]. Such a bijection has been given by de Medicis and Viennot [5] but their construction is indirect in the sense that it decomposes a permutation in two involutions, then goes through the construction and the fusion of two Hermite histories. Another interest of our construction is that it gives a link between subdivided Laguerre histories and tree-like tableaux [1], which are a new presentation of permutation tableaux [8] or alternative tableaux [10]. The link lies in the insertion algorithm used on both classes of objects and whose key ingredient is the notion of ribbon. For this reason, the central objects of this paper are tableaux called Dyck tableaux whose natural reading in terms of words gives subdivided Laguerre histories. Although the original construction is not recursive, we are able to easily describe relevant statistics (generalized patterns, shape, (RL/LR)-(minima/maxima)) because of the recursive structure given by the insertion procedure. When talking about relevant statistics, we have in mind the long-term challenging motivation of this work: build new objects in order to give a new, and if possible simpler,