The parallel TASEP, fixed particle number and weighted Motzkin paths

In this paper the totally asymmetric exclusion process (TASEP) with parallel update on an open lattice of size $L$ is considered in the maximum-current region. A formal expression for the generating function for the weight of configurations with $N$ particles is given. Further an interpretation in terms of $(u,l,d)$-colored weighted Motzkin paths is presented. Using previous results (Woelki and Schreckenberg 2009 {\it J. Stat. Mech} P05014, Woelki 2010 {\it Cellular Automata}, pp 637-645) the generating function is compared with the one for a possible 2nd-class particle dynamics for the parallel TASEP. It is shown that both become physically equivalent in the thermodynamic limit.