How to deduce a formula (see below) for number of conjugacy classes in $\operatorname{GL}_n(F_2)$? (More generally F_q) ? Is there some description of conjugacy classes or we just know how many of them but do not know how to describe them ?

Can someone send me a paper, please ?: "Pairs of commuting matrices over a finite field" Walter Feit and N. J. Fine

**FORMULA from OEIS**
The number $a(n)$ of conjugacy classes in the group $\operatorname{GL}(n, q)$ is the coefficient of $t^n$ in the infinite product: product $k=1, 2, ... (1-t^k)/(1-qt^k)$ - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001.

## Simple observations

Clearly if characteristic polynom is different then matrices are not conjugated. For each char. pol. it is easy to give a matrix with such char. pol. so we get (q-1)q^{n-1} possibilities at least.

But if char. pol has a roots with multiplicities - then we may have several Jordan cells and several conjugacy classes with same characteristic polynomial. So it is not clear for me how to count them.

Diagonal elements of corresponding Jordan cells may not live in F_q but in some alg. extentsion of it, these makes me completely puzzled - is it possible to control this ? Hardly...

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Enumerative Combinatorics, vol. 1, second ed. See math.mit.edu/~rstan/ec/ec1.pdf. For fixed $n$, the number of conjugacy classes is a polynomial in $q$ of the form $q^n-q^{\lfloor (n-1)/2\rfloor}+O(q^{\lfloor (n-1)/2\rfloor -1})$. Additional properties of this polynomial appear in this exercise. $\endgroup$2more comments