5 asymptotic formula corrected

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 $$\sum_{n\geq 0}\omega^\ast(n,q)x^n = \prod_{j\geq 1} \frac{1-x^j}{1-qx^j}$$ $$\omega^\ast(n,q) = q^n-q^m- q^{m-1}-q^{m-2}-\cdots -q^{2\lfloor (n+5)/6\rfloor}+O(q^{\lfloor (n+5)/6\rfloor-1}), q^{\lfloor n/3\rfloor}+O(q^{\lfloor n/3\rfloor-1}),$$ where $m=\lfloor (n-1)/2\rfloor$.

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/ch1.pdfhttp://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 $$\sum_{n\geq 0}\omega^\ast(n,q)x^n = \prod_{j\geq 1} \frac{1-x^j}{1-qx^j}$$ $$\omega^\ast(n,q) = q^n-q^m- q^{m-1}-q^{m-2}-\cdots -q^{2\lfloor (n+5)/6\rfloor}+O(q^{\lfloor (n+5)/6\rfloor-1}),$$ where $m=\lfloor (n-1)/2\rfloor$.
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/ch1.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 $$\sum_{n\geq 0}\omega^\ast(n,q)x^n = \prod_{j\geq 1} \frac{1-x^j}{1-qx^j}$$ $$\omega^\ast(n,q) = q^n-q^m- q^{m-1}-q^{m-2}-\cdots -q^{2\lfloor (n+5)/6\rfloor}+O(q^{\lfloor (n+5)/6\rfloor-1}),$$ where $m=\lfloor (n-1)/2\rfloor$.