  In this paper we prove a new combinatorial formula for the modified Macdonald
polynomials $\widetilde{H}_\lambda(X;q,t)$, motivated by connections to the
theory of interacting particle systems from statistical mechanics. The formula
involves a new statistic called queue inversions on fillings of tableaux. This
statistic is closely related to the multiline queues which were recently used
to give a formula for the Macdonald polynomials $P_\lambda(X;q,t)$. In the case
$q=1$ and $X=(1,1,\dots,1)$, that formula had also been shown to compute
stationary probabilities for a particle system known as the multispecies ASEP
on a ring, and it is natural to ask whether a similar connection exists between
the modified Macdonald polynomials and a suitable statistical mechanics model.
In a sequel to this work, we demonstrate such a connection, showing that the
stationary probabilities of the multispecies totally asymmetric zero-range
process (mTAZRP) on a ring can be computed using tableaux formulas with the
queue inversion statistic. This connection extends to arbitrary $X=(x_1,\dots,
x_n)$; the $x_i$ play the role of site-dependent jump rates for the mTAZRP.

Combinatorics