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 inhomogeneous rates given by parameters. We introduce a new object that we call 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 multiline queues of Ferrari and Martin. Using the tableaux, we obtain a formula for the stationary probabilities of the inhomogeneous two-species 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.