This is a question more from combinatorics although its background is in linear algebra.
Let $A$ be a real matrix of dimension $n \times n$. We know that in general the sequence of matrices $A, A^2, A^3, \dots $ can have decreasing rank - in extreme case if for some $k$ we have $A^k=0$ then the matrix is (called) nilpotent and the patterns of transforming the rank from an initial value to $0$ can be very different, in general we could write such rank pattern $ m_1\rightarrow m_2\rightarrow \dots \rightarrow m_k=0 $.
Of course if a matrix is of full rank the exponentiation preserves rank: we have always $ n \rightarrow n$ (single possible pattern).
When the rank is less than $n$ it is also possible that exponentiation doesn't change the rank.
But the situation when the rank is decreasing from some value of $m_1$ to $m_k$ is possible in many different ways. I'm interested in the number of different ways how it could be done. It obviously depends on the possible Jordan Normal Forms for $n \times n$ matrix.
- What is the number formula for these forms if the only thing which is used in classification here is the way the rank decreases?
(what corresponds to unique possible sequences of changes in rank for n-dimensional matrix $ m_1\rightarrow m_2\rightarrow \dots \rightarrow m_k$)