  Let $B$ be an $n \times n$ nilpotent matrix with entries in an infinite field
$\k$. 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 $\lambda_U(P)$.
