I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation.
For example, if a Jordan form consists of a single cell $2 \times 2$ $J=\begin{pmatrix} \lambda_0 &1\\\ 0&\lambda_0\end{pmatrix},$ by small perturbation $\begin{pmatrix} 0&0\\\ 0&\varepsilon\end{pmatrix}$, we can only get the matrix $\begin{pmatrix} \lambda_0 &0\\\ 0&\lambda_1\end{pmatrix}$ i.e. only two variants are possible.
If the Jordan form consists of a single cell 3x3, there may be such cases: $J=\begin{pmatrix} \lambda_0 &1&0\\\ 0&\lambda_0&1\\\ 0&0&\lambda_0\end{pmatrix},$ $J+ \begin{pmatrix} 0 &0&0\\\ 0&0&0\\\ 0&0&\varepsilon\end{pmatrix}\sim\begin{pmatrix} \lambda_0 &1&0\\\ 0&\lambda_0&0\\\ 0&0&\lambda_1\end{pmatrix},$ $J+ \begin{pmatrix} 0 &0&0\\\ 0&\varepsilon_1&0\\\ 0&0&\varepsilon_2\end{pmatrix}\sim\begin{pmatrix} \lambda_0 &0&0\\\ 0&\lambda_1&0\\\ 0&0&\lambda_2\end{pmatrix}.$
i.e. only three variants are possible.
I think I proved that if the Jordan form consists of a single cell $m \times m$, then the number of variants equal to $p(m)$ (see http://en.wikipedia.org/wiki/Partition_%28number_theory%29).
It seems to me that these results have been obtained by someone, but I can not find them.
We are working over $\mathbb{C}$.
Able to show that the number of possible Jordan forms of the matrix $n\times n$ gives the number $a(n)$ (see http://oeis.org/A001970). Remains to determine how many Jordan forms can not be receive from this.