Weighted Staircase Tableaux, Asymmetric Exclusion Process, and Eulerian Type Recurrences

We consider a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years. We will be interested in a general model of staircase tableaux in which symbols that appear in staircase tableaux may have arbitrary positive weights. Under this general model we derive a number of results concerning the limiting laws for the number of appearances of symbols in a random staircase tableaux. One advantage of our generality is that we may let the weights approach extreme values of zero or infinity, which covers further special cases appearing earlier in the literature. One of the main tools we use are generating functions of the parameters of interests. This leads us to a two–parameter family of polynomials. Specific values of the parameters cover a number of special cases analyzed earlier in the literature including the classical Eulerian polynomials.