  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
$A_n$ or $C_n$, we provide a combinatorial explanation of this coincidence by
constructing a suitable bijection. We also construct another general bijection
between $Ab(g)$ and the subsets of the Dynkin diagram, which is based on the
theory developed by Peterson and Kostant.
