  Let $k$ be an infinite field. Fix a Jordan nilpotent $n$ by $n$ matrix $B =
J_P$ with entries in $k$ and associated Jordan type $P$. Let $Q(P)$ be the
Jordan type of a generic nilpotent matrix commuting with $B$. In this paper, we
use the combinatorics of a poset associated to the partition $P$, to give an
explicit formula for the smallest part of $Q(P)$, which is independent of the
characteristic of $k$. This, in particular, leads to a complete description of
$Q(P)$ when it has at most three parts.
