Toric tableaux and the inhomogeneous two-species TASEP on a ring

The inhomogeneous two-species TASEP on a ring is an exclusion process that describes particles of different species hopping clockwise on a ring with parameters giving the hopping rates for different species. We introduce a combinatorial object that we call \emph{toric rhombic alternative tableaux}, which are certain fillings of tableaux on a triangular lattice tiled with rhombi, and are in bijection with the well-studied \emph{multiline queues} of Ferrari and Martin. Using the tableaux, we obtain a formula for the stationary probabilities of this TASEP, which specializes to results of Ayyer and Linusson. We obtain, in addition, an explicit determinantal formula for these probabilities, and define a Markov chain on the tableaux that projects to the two-species TASEP on a ring.