We show that the known matrix representations of the stationary state algebra of the Asymmetric Simple Exclusion Process (ASEP) can be interpreted combinatorially as various weighted lattice paths. This interpretation enables us to use the constant term method (CTM) and bijective combinatorial methods to express many forms of the ASEP normalisation factor in terms of Ballot numbers. One particular lattice path representation shows that the coefficients in the recurrence relation for the ASEP correlation functions are also Ballot numbers. Additionally, the CTM has a strong combinatorial connection which leads to a new ``canonical lattice path representation and to the ``omega-expansion which provides a uniform approach to computing the asymptotic behaviour in the various phases of the ASEP. The path representations enable the ASEP normalisation factor to be seen as the partition function of a more general polymer chain model having a two parameter interaction with a surface.
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