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

Let kk be an infinite field. Fix a Jordan nilpotent n×nn×n Jordan matrix B   with entries in kk and associated Jordan type P  . Let Q(P)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)Q(P), which is independent of the characteristic of kk. This, in particular, leads to a complete description of Q(P)Q(P) when it has at most three parts.