A determinant-like formula for the Kostka numbers

Abstract

Young tableaux are ubiquitous in various branches of mathematics. There are two counting formulas for standard Young tableaux. The first involves a determinant and goes back to Frobenius and Young, and the second is the hook formula by Frame, Robinson and Thrall. We present a generalization of the determinant formula for semistandard Young tableaux of given shape and of given content. Our counting formula -- though not a determinant -- is a generalization of the determinant formula by Frobenius and Young.