Rhombic alternative tableaux and assembl\'ees of permutations

In this paper, we introduce the rhombic alternative tableaux, whose weight generating functions provide combinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there are two species of particles, one heavy and one light, hopping right and left on a one-dimensional finite lattice with open boundaries. Parameters $\alpha$, $\beta$, and $q$ describe the hopping probabilities. The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assembl\'ees of permutations. We describe a bijection between the rhombic alternative tableaux and these assembl\'ees. We also provide an insertion algorithm that gives a weight generating function for the assembl\'ees. Combined, these results give a bijective proof for the weight generating function for the rhombic alternative tableaux, which is also the partition function of the two-species ASEP at $q=1$.