We describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make this connection explicit. We give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ in terms of a queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle; moreover, the queue inversion statistic is naturally related to the dynamics of the ASEP. Our tableaux are in bijection with Martin's multiline queues, from which we obtain an alternative multiline queue formula for $P_{\lambda}$. The new formula arises from the plethystic correspondence between the classical and modified Macdonald polynomials, which is closely related to fusion in the setting of integrable systems which connects the ASEP to the TAZRP.
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