  We note that a tridiagonal matrix representation of the algebra of the
partially asymmetric exclusion process (PASEP) lends itself to interpretation
as the transfer matrix for weighted Motzkin lattice paths. A continued fraction
("J-Fraction") representation of the lattice path generating function is
particularly well suited to discussing the PASEP, for which the paths have
height dependent weights. We show that this not only allows a succinct
derivation of the normalisation and correlation lengths of the PASEP, but also
reveals how finite-dimensional representations of the PASEP algebra, valid only
along special lines in the phase diagram, relate to the general solution that
requires an infinite-dimensional representation.
