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.



Finite nilpotent orbits: GL(n,q)-conjugacy classes and a partial order on partitions
I think the right way is to factor $n\_\lambda(q)$ in general. In particular, it is obviously a quotient of the order of $GL\_n(q)$. The formula for the order of $GL\_n(q)$ does not have very many prime factors: just $q$ and the first $n$ cyclotomic polynomials.


One could consider an alternate question, the order of the centralizer of $J$ in $GL(n,\mathbb F\_q)$. This gives the reverse of the partial order, since the divisibility relation is reversed. The centralizer is the automorphism group of the corresponding $\mathbb F\_q[x]/x^n$-module, $M$. The subgroup that fixes $M/x$ is a $q$-group, since it is unipotent. Its quotient is a product of copies of $GL(k,\mathbb F\_q)$: one for each type of block, with $k$ equal to the number of that block that appears. 


The number of times that the $k$th elementary cyclotomic polynomial appears in the size of the centralizer is just the sum over all sizes $n$ of the floor of $a\_n/k$, where $a\_n$ is the number of blocks of size $n$.


This function behaves very erratically, so we can conclude that the partial order behaves erratically as well.


