# Mathematics Colloquia and Seminars

The partially asymmetric exclusion process (PASEP) is an important and well-studied non-equilibrium model from statistical physics, in which particles hop left and right on a finite one-dimensional lattice with open boundaries. Typically the model has 3 hopping parameters (governing the rates at which particles enter and exit the lattice, and hop left and right). The two-species PASEP is a generalization of the PASEP in which there are two types of particles, one heavy'' and one light''. Much past work was devoted to finding combinatorial formulas for the steady state probabilities of the original PASEP. In this talk I will present our recent work on the combinatorics of the two-species PASEP. Together with Viennot, we introduced rhombic alternative tableaux,'' which are fillings of certain rhombic tilings, and used them to provide combinatorial formulas for the two-species PASEP. We also introduced k-rhombic alternative tableaux,'' with analogous combinatorial formulas for a more general k-species PASEP. Finally, in recent joint work with Corteel and Williams, we introduced rhombic staircase tableaux'', and provided combinatorial formulas for the more general 5-parameter two-species PASEP. This talk will be accessible to graduate students in combinatorics.