Generalization of the Gale–Ryser Theorem

We prove an inequality for the Kostka–Foulkes polynomials Kλ,μ(q) and give a criteria for the existence of a unique configuration of the given type (λ, μ). As a corollary, we obtain a nontrivial lower bound for the Kostka numbers which is a generalization the Gale–Ryser theorem on an existence of a (0,1)-matrix with given sums of rows and columns. A new proof of the Berenstein–Zelevinsky weight-multiplicity-one criteria is given.