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.
