I am trying to understand how we count the number of possible rational canonical forms for a matrix with given minimal polynomial up to the order of companion blocks. The book we are using is sort of abstract not great on the computational side (Roman, Advanced Linear Algebra) and I want to see if my "intuitive understanding" maps back correctly to theory.
So, let $A\in \mathbb{F}^n$ be a matrix and suppose it has minimal polynomial $q$ and suppose it has degree $m<n$ (for else the question is trivial). Further, let $p$ be the characteristic polynomial of $A$. Now, we know that $q|p$ and any root of $p$ is a root $q$, so if for instance $q=(x+1)^3(x-1)^2$ then clearly $p=(x+1)^3(x-1)^2 \times (x+1)^k(x-1)^j$ with $j+k=n-m$.
Hence, the number of possible characteristic polynomials is easily determined by solving and integer equation of the type $j+k=n-m$. However, the same characteristic polynomial can give rise to different rational canonical forms.
Now, what I don't understand is what companion blocks are "allowed" for each characteristic polynomial? I mean, $(x-1),(x-1)$ clearly gives a different block than $(x-1)^2$ so count these as different possible forms, but what powers of $x-1$ are allowed to be in the form? What $(x-1)^k$ blocks can I actually count as being an allowed form and in particular for what $k$ (given the example above)? And why?
EDIT: similarly, for what $j$ are $(x+1)^j$ blocks allowed?
EDIT2: After thinking further on it, I think the maximal admissible $k$ should be $k=2$ and similarly $j=3$. Correct? This stems from the primary cycle decomposition for modules.