  Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An
ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The
generators of an ad-nilpotent ideal give rise to an antichain in the poset of
positive roots, and the whole theory can be expressed in a combinatorial
fashion, in terms of antichains. The aim of this paper is to present a
refinement of the enumerative theory of ad-nilpotent ideals for the case in
which $\g$ has roots of different length. An antichain is called short, if it
consists of short roots. We obtain, for short antichains, analogues of all
results known for the usual antichains.
