Paths and Kostka–Macdonald polynomials

We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the $q$-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra $\mathfrak{gl}(n)$. As an application, we give an elementary proof of the special case $t=1$ of the Haglund–Haiman–Loehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the so-called energy statistics.