  We review various combinatorial interpretations and mappings of
stationary-state probabilities of the totally asymmetric, partially asymmetric
and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In
these steady states, the statistical weight of a configuration is determined
from a matrix product, which can be written explicitly in terms of generalised
ladder operators. This lends a natural association to the enumeration of random
walks with certain properties.
  Specifically, there is a one-to-many mapping of steady-state configurations
to a larger state space of discrete paths, which themselves map to an even
larger state space of number permutations. It is often the case that the
configuration weights in the extended space are of a relatively simple form
(e.g., a Boltzmann-like distribution). Meanwhile, various physical properties
of the nonequilibrium steady state - such as the entropy - can be interpreted
in terms of how this larger state space has been partitioned.
  These mappings sometimes allow physical results to be derived very simply,
and conversely the physical approach allows some new combinatorial problems to
be solved. This work brings together results and observations scattered in the
combinatorics and statistical physics literature, and also presents new
results. The review is pitched at statistical physicists who, though not
professional combinatorialists, are competent and enthusiastic amateurs.
