Stationary probabilities of the multispecies TAZRP and modified Macdonald polynomials: I

Recently, a formula for the Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a particle model from statistical mechanics called the multispecies ASEP on a ring. It is natural to ask whether the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative. In this paper we give a new formula for $\widetilde{H}_{\lambda}(X;q,t)$ in terms of fillings of tableaux called polyqueue tableaux. In the upcoming sequel to this paper, we show that polyqueue tableaux also compute probabilities of the multispecies totally asymmetric zero range process (mTAZRP) on a ring, and that $\widetilde{H}_{\lambda}(X;1,t)$ is equal to the partition function of the mTAZRP.