  It is well-known that a nilpotent n by n matrix B is determined up to
conjugacy by a partition of n formed by the sizes of the Jordan blocks of B. We
call this partition the Jordan type of B. We obtain partial results on the
following problem: for any partition P of n describe the type Q(P) of a generic
nilpotent matrix commuting with a given nilpotent matrix of type P. A
conjectural description for Q(P) was given by P. Oblak and restated by L.
Khatami. In this paper we prove "half" of this conjecture by showing that this
conjectural type is less than or equal to Q(P) in the dominance order on
partitions.
