Titre: Combinatoire des tableaux et polynômes orthogonaux

• Permutation tableaux are new objects that come from the enumeration of the totally positive Grassmannian cells [11, 15]. A permutation tableau [12] is a Ferrers diagram together with a filling of the cells with 0’s and 1’s such that the following properties hold: Each column contains at least one 1 and there is no 0 which has a 1 above it in the same column and a 1 to its left in the same row. They are called permutation tableaux because they are in bijection with permutations [12]. Different statistics on permutation tableaux were defined in [5, 12] and they have counter part in permutations. Surprisingly they are also connected to a statistical physics model called the Partially ASymmetric Exclusion Process (PASEP) [4, 5]. This connection was made thanks to previous work by Duchi and Schaeffer [6]. In particular, it was shown in [4] that the probability of finding the PASEP in configuration tau in the steady state is the probability generating function of the tableau of shape tau, according to the number of superfluous ones, the number of ones in the first row and the number of unrestricted rows minus one. Moreover, the partition function Zn for the PASEP is equal to the weight-generating function for all permutation tableaux of length n+1. In [5], we defined a Markov chain on the permutation tableaux that gave a combinatorial proof of this result. Moreover this Partially ASymmetric Exclusion Process has more generalizations than the ones studied in [4, 5].