  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. Additionally, particles may enter from the left
with probability $\alpha$ and exit from the right with probability $\beta$.
  In this paper we prove a close connection between the PASEP and the
combinatorics of permutation tableaux. (These tableaux come indirectly from the
totally nonnegative part of the Grassmannian, via work of Postnikov, and were
studied in a paper of Steingrimsson and the second author.) Namely, we prove
that in the long time limit, the probability that the PASEP is in a particular
configuration $\tau$ is essentially the generating function for permutation
tableaux of shape $\lambda(\tau)$ enumerated according to three statistics. The
proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on
the {\it matrix ansatz} for the PASEP model.
  As an application, we prove some monotonicity results for the PASEP. We also
derive some enumerative consequences for permutations enumerated according to
various statistics such as weak excedence set, descent set, crossings, and
occurences of generalized patterns.
