  We present a determinantal formula for the steady state probability of each
state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open
boundaries, a 1D particle model that has been studied extensively and displays
rich combinatorial structure. These steady state probabilities are computed by
the enumeration of Catalan tableaux, which are certain Young diagrams filled
with $\alpha$'s and $\beta$'s that satisfy some conditions on the rows and
columns. We construct a bijection from the Catalan tableaux to weighted lattice
paths on a Young diagram, and from this we enumerate the paths with a
determinantal formula, building upon a formula of Narayana that counts
unweighted lattice paths on a Young diagram. Finally, we provide a formula for
the enumeration of Catalan tableaux that satisfy a given condition on the rows,
which corresponds to the steady state probability that in the TASEP on a
lattice with $n$ sites, precisely $k$ of the sites are occupied by particles.
This formula is an $\alpha\ /\ \beta$ generalization of the Narayana numbers.
