  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 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.
