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

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Let k be an infinite field. Fix a Jordan nilpotent n by n matrix B = Jₚ 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.