  Inhomogeneous lattice paths are introduced as ordered sequences of
rectangular Young tableaux thereby generalizing recent work on the Kostka
polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon.
Motivated by these works and by Kashiwara's theory of crystal bases we define a
statistic on paths yielding two novel classes of polynomials. One of these
provides a generalization of the Kostka polynomials while the other, which we
name the A$_{n-1}$ supernomial, is a $q$-deformation of the expansion
coefficients of products of Schur polynomials. Many well-known results for
Kostka polynomials are extended leading to representations of our polynomials
in terms of a charge statistic on Littlewood-Richardson tableaux and in terms
of fermionic configuration sums. Several identities for the generalized Kostka
polynomials and the A$_{n-1}$ supernomials are proven or conjectured. Finally,
a connection between the supernomials and Bailey's lemma is made.
