A combinatorial derivation of the PASEP stationary state

We give a combinatorial derivation and interpretation of the algebra associated with the stationary distribution of the partially asymmetric exclusion process. Our derivation works by constructing a larger Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original chain and that the stationary distribution on this subset is simply related to that of the original chain. This derivation allows us to find expressions for the normalisation using both recurrences and path models. These results exhibit classical combinatorial numbers such as n!, 2n and the Catalan numbers