Commuting Nilpotent Matrices and Oblak’s Proposed Formula

Abstract

Let N(k) be the set of all n ? n nilpotent matrices with entries in k. A matrix M in N(k) is determined up to conjugacy by its Jordan type P, a partition of n giving the sizes of the Jordan blocks of B. What is the largest Jordan type Q(P) of a nilpotent matrix A commuting with B? P. Oblak has a conjecture for Q(P) that is true when Q(P) has no more than 3 parts (P. Oblak, L. Khatami). She proposed a formula for the number of P satisfying Q(P)=Q when Q = (u,u-r) has two parts, and she showed it for r