The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the
probability of hopping left is q times the probability of hopping right. In this paper we prove a close connection between the PASEP model and the combinatorics of
permutation tableaux (certain 0-1 tableaux introduced in a previous paper with Steingrimsson). Namely, we prove that in the long time limit, the probability that the PASEP model is in a particular configuration
tau is essentially the weight
generating function for permutation tableaux of shape lambda(tau). The proof of this result uses a result of Derrida et al on the matrix ansatz for the PASEP. We derive a number of enumerative consequences of the connection between the PASEP model and permutation tableaux. One consequence is a generating function for the following (equidistributed) objects: the partition function for the PASEP model; permutation tableaux of length n+1, enumerated according to weight;
permutations in S_{n+1}, enumerated according to crossings; permutations in S_{n+1}, enumerated according to occurrences of the generalized pattern 2-31. Another consequence is a generating function for the subset of the above objects which is specified by fixing (respectively) a configuration tau, a shape lambda(tau), a weak excedence set W(tau), or a descent set D(tau). Note that the equidistribution of permutation tableaux and permutations was proved in a previous paper of Steingrimsson and the author