  Kostka-Foulkes polynomials are Lusztig's $q$-analogues of weight
multiplicities for irreducible representations of semisimple Lie algebras. It
has long been known that these polynomials have non-negative coefficients. A
statistic on semistandard Young tableaux with partition content, called
\textit{charge}, was used to give a combinatorial formula exhibiting this fact
in type $A$. Defining a charge statistic beyond type $A$ has been a
long-standing problem. Here, we take a completely new approach based on the
definition of Kostka-Foulkes polynomials as an alternating sum over Kostant
partitions, which can be thought of as formal sums of positive roots. We use a
sign-reversing involution to obtain a positive expansion, in which the relevant
statistic is simply the number of parts in the Kostant partitions. The hope is
that the simplicity of this new crystal-like model will naturally extend to
other classical types.
