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 α's and β'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 α/β generalization of the Narayana numbers.
Mandelshtam, O. (2015). A determinantal formula for Catalan tableaux and TASEP probabilities. Journal of Combinatorial Theory. Series A, 132, 120–141. https://doi.org/10.1016/j.jcta.2014.12.005