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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 theJordan type of a generic nilpotent matrix commuting with $B$. In this paper, weuse the combinatorics of a poset associated to the partition $P$, to give anexplicit formula for the smallest part of $Q(P)$, which is independent of thecharacteristic of $k$. This, in particular, leads to a complete description of$Q(P)$ when it has at most three parts.

The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix