TITLE: Combinatorial Techniques for Counting Conjugacy Classes
QUESTION [24 upvotes]: The number of conjugacy classes in $S_n$ is given by the number of partitions of $n$. Do other families of finite groups have a highly combinatorial structure to their number of conjugacy classes? For example, how much is known about conjugacy classes in $A_n$?

REPLY [14 votes]: For an explicit formula for the size of the conjugacy classes of GL$(n,q)$ (going back to Frobenius and Philip Hall), see equation (1.107) on page 92 of http://math.mit.edu/~rstan/ec/ec1.pdf. (There are formulas for the quantities appearing in (1.107) earlier in the text and in (1.108).)
As for the number $\omega^\ast(n,q)$ of conjugacy classes, see Exercise
1.190 on page 156 of the above reference. In particular, for fixed $n$,
$\omega^\ast(n,q)$ is a polynomial in $q$ satisfying 
  $$ \sum_{n\geq 0}\omega^\ast(n,q)x^n = \prod_{j\geq 1}
    \frac{1-x^j}{1-qx^j} $$
  $$ \omega^\ast(n,q) = q^n-q^m- q^{m-1}-q^{m-2}-\cdots
    -q^{\lfloor n/3\rfloor}+O(q^{\lfloor n/3\rfloor-1}), $$
where $m=\lfloor (n-1)/2\rfloor$.