In this paper we study a relatively new combinatorial object called staircase tableaux. Staircase tableaux were introduced by Corteel and Williams in the connection with Asymmetric Exclusion Process and has since found interesting connections with Askey-Wilson polynomials. We develop a probabilistic approach that allows us to analyze several parameters of a randomly chosen staircase tableau of a given size. In particular, we obtain limiting distributions for statistics associated with appearances of Greek letters in staircase tableaux.Key words and phrases: asymmetric exclusion process, asymptotic normality, staircase tableaux. introductionAn interesting combinatorial structure, called staircase tableaux, was introduced in recent work of Corteel and Williams [11,12]. Staircase tableaux are related to the asymmetric exclusion process on an one-dimensional lattice with open boundaries, the ASEP. This is an important and heavily studied particle model in statistical mechanics (we refer to [12] for some background information on several versions of that model and their applications and connections to other branches of science). The study of the generating function of the staircase tableau has given a combinatorial formula for the steady state probability of the ASEP. Explicit expressions for the steady state probabilities were first given in [15]. In their work [12,11] Corteel and Williams used staircase tableaux to give a combinatorial formula for the moments of the (weight function of the) Askey-Wilson polynomials; for a follow-up work see [7].

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International audience We give a simple bijection between some staircase tableaux and tables of inversion. Some nice properties of the bijection allows us to define some q-Eulerian polynomials related to the staircase tableaux. We also give a combinatorial interpretation of these q-Eulerian polynomials in terms of permutations. Nous proposons une bijection simple entre certains tableaux escalier et les tables d'inversion. Cette bijection nous permet de montrer que les statistiques Euleriennes et Mahoniennes sont naturelles sur les tableaux escalier. Nous définissons des polynômes q-Eulériens et en donnons une interprétation combinatoire.

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