The poset of the nilpotent commutator of a nilpotent matrix

Let $B$ be an $n\times n$ nilpotent matrix with entries in an infinite field $\text{k}$. Assume that $B$ is in Jordan canonical form with the associated Jordan block partition $P$. In this paper, we study a poset ${\mathcal{D}}_P$ associated to the nilpotent commutator of $B$ and a certain partition of $n$, denoted by ${\lambda }_U(P)$, defined in terms of the lengths of unions of special chains in ${\mathcal{D}}_P$. Polona Oblak associated to a given partition $P$ another partition $Ob(P)$ resulting from a recursive process. She conjectured that $Ob(P)$ is the same as the Jordan partition $Q(P)$ of a generic element of the nilpotent commutator of $B$. Roberta Basili, Anthony Iarrobino and the author later generalized the process introduced by Oblak. In this paper we show that all such processes result in the partition ${\lambda }_U(P)$.