Segre Characteristic Equivalence

Given only the dimension, $n$, of a square matrix $A\in M(n,\mathbb{C})$, how many Segre Characteristic equivalent matrices are there? Jordan Normal Form Theorem states that any linear operator over $\mathbb{C}$ is similar to a matrix in Jordan Normal Form. As such, this is a question of counting the number of possible Jordan Normal Forms for a given dimension. So, equivalently, how many Jordan Normal Forms can an $n\times n$ matrix possibly have?