  Based on the matrix ansatz of Derrida, Evans, Hakim and Pasquier, we prensent
a new way of computing the stationary probability of a state of the asym-
metric simple exclusion process (ASEP). Through an insertion algorithm over
staircase tableaux, we give a combinatorial proof to the current interpretation
of the ASEP by these tableaux of Corteel and Williams. The insertion algorithm
induces a recursive structure which implies nice factorised formulas for the
generating polynomials of staircase tableaux, as well as a bijection with some
coloured inversion tables. In addi- tion, we adapt the insertion algorithm to
the case of type B symmetric tableaux and we define a new matrix ansatz
compatible with it.
