Inhomogeneous lattice paths, generalized Kostka polynomials and A(n-1) supernomials

Inhomogeneous lattice paths are introduced as ordered sequences of rectangu lar Young tableaux thereby generalizing recent work on the Kostka polynomia ls 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 stati stic on paths yielding two novel classes of polynomials. One of these provi des a generalization of the Kostka polynomials, while the other, which we n ame the A(n-1) supernomial, is a q-deformation of the expansion coefficient s of products of Schur polynomials. Many well-known results for Kostka poly nomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood-Richardson tableaux and in terms of fe rmionic configuration sums. Several identities for the generalized Kostka p olynomials and the A(n-1) supernomials are proven or conjectured. Finally, a connection between the supernomials and Bailey's lemma is made.