I have a question regarding a partial order $<$ on the set ${\rm Part}(n)$ of partitions of $n$. Given $\lambda=(\lambda_1,\lambda_2,\ldots)\in{\rm Part}(n)$ with $\sum_{i\geq1} \lambda_i=n$ and $\lambda_1\geq\lambda_2\geq\cdots\geq0$, let $J_\lambda$ denote the $n\times n$ block diagonal matrix $\bigoplus_{i\geq1}J_{\lambda_i}$. For example, $J_{(3,2,1)}=\left(\begin{smallmatrix}0&1&0&&&\\0&0&1&&&\\0&0&0&&&\\&&&0&1&\\&&&0&0&\\&&&&&0\end{smallmatrix}\right)$. Consider the ${\rm GL}(n,F)$-conjugacy classes of the set ${\rm M}(n,F)$ of all $n\times n$ matrices over a field $F$. A nilpotent matrix $X\in{\rm M}(n,F)$ lies in a conjugacy classes $\mathcal{O}_\lambda:=J_\lambda^{{\rm GL}(n,F)}$ for a unique $\lambda\in{\rm Part}(n)$. (Nilpotent means $X^n=0$.)

If $F=\mathbb{F}_q$ is a finite field, then set
$n_\lambda:=|J_\lambda^{{\rm GL}(n,q)}|$. A formula for $n_\lambda$
is given in Fulman, Cycle indices for finite classical groups. It turns out
that $n_\lambda=n_\lambda(q)$ is a polynomial in $q$ with integer coefficients.
Define a partial order $<$ on ${\rm Part}(n)$ as follows:
$\lambda<\mu$ if and only if $n_\lambda(q)$ divides $n_\mu(q)$.
I call this the *divisibility* partial order.

When $F$ is the complex field $\mathbb{C}$, define $\lambda\triangleleft\mu$
if $\overline{\mathcal{O}_\lambda}\subset\overline{\mathcal{O}_\mu}$ where
$\overline{\mathcal{O}_\lambda}$ denotes the Zariski closure of
$\mathcal{O}_\lambda$. It is shown in Collingwood and McGovern, Nilpotent
orbits of semisimple Lie algebras, pp 93--95, that $\triangleleft$ is the
*dominance* partial order on ${\rm Part}(n)$. That is,
$\lambda\triangleleft\mu$ if and only
if $\sum_{i=1}^{k-1}\lambda_i=\sum_{i=1}^{k-1}\mu_i$ and $\lambda_k<\mu_k$
for some $k\geq1$.

If $n\leq5$, then the partial orders $<$ and $\triangleleft$ are identical and are total orders.
However, when $n=6$ the partition $(3,2,1)$ of 6 has three partitions
*divisibility larger*, and has five partitions *dominance larger*.

Does anyone have any insight into divisibility partial order? or know of
its appearance in the literature? (I have not found a reference to $<$ in
Roger Carter's book Finite groups of Lie type: conjugacy classes and
complex characters, but $\triangleleft$ appears in 5.5 and 5.11.)
For specific $\lambda$, I can (theoretically) factor $n_\lambda(q)$ and so
can determined whether $\lambda<\mu$ for *specific* $\lambda$ and $\mu$, but
I have few global results.