Abelian ideals of a Borel subalgebra and subsets of the Dynkin diagram

Let g be a simple Lie algebra and Ab(g) the set of abelian ideals of a Borel subalgebra of g. In this note, an interesting connection between Ab(g) and the subsets of the Dynkin diagram of g is discussed. We notice that the number of abelian ideals with k generators equals the number of subsets of the Dynkin diagram with k connected components. For g of type An or Cn, we provide a combinatorial explanation of this coincidence by constructing a suitable bijection. We also construct a general bijection between Ab(g) and the subsets of the Dynkin diagram, which is based on the theory developed by Peterson and Kostant.