Irreducible components of Hilbert scheme of points on non-reduced curves

We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions of $n$; all have dimension $n$; and their multiplicities are given as a polynomial of the parts of the corresponding partitions.