Link between Irreducible Factors and Prime Factors (or Cycles of a Permutation) In "Anatomy of Integers and Permutations", http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf, Granville gives a calibration of cycles of a permutation and prime factors of an integer. "We know roughly one out of evert $\log x$ integers up to $x$ is a prime, and that exactly one in every $N$ permutations on $N$ elements is a cycle, so we could try to calibrate by replacing $N$ when we measure the anatomy of a permutation with $\log x$ when we measure the anatomy of an integer."
Is there also a useful link between irreducible factors of a monic polynomial of degree $n$ over $F_q[T]$ and either cycles or prime factors?
A naive guess would be to say that the link between irreducible factors and cycles is trivial due to the similarity of the following theorems:
Prime Number Theorem for Permutations: A randomly selected permutation of $S_n$ will be an $n$-cycle with probability exactly $1/n$.
Prime Number Theorem for Monic Polynomials: The probability that a random monic polynomial $f\in F_q[T]$ of degree $n$ will be irreducible with probability $\approx 1/n$ when $q$ is fixed and $n$ is large (or if $n$ is fixed and characteristic of $F_q$ is large). 
 A: There is such a link and it is in fact much better understood in the polynomial setting. Namely, if $f$ is a separable polynomial over a finite field, then the action of the Frobenius element on the roots of $f$ gives rise to a permutation whose cycle structure corresponds to the factorization type of $f$. 
Then, combinatorial arguments and/or algebraic arguments (such as the Chebotarev Density Theorem) tell us that the distribution of those permutations as $f$ varies over degree $n$ polynomials is roughly the uniform distribution on $S_n$ (as $q$ grows they get closer).
See Terry Tao's post on the subject, which refers to Granville's paper as well:
https://terrytao.wordpress.com/2015/07/15/cycles-of-a-random-permutation-and-irreducible-factors-of-a-random-polynomial/
A: I wrote a blog post on this here. The basic result is that for fixed $k$ and $n$, as $q \to \infty$ the joint distribution of irreducible factors of degrees $1$ through $k$ of a random monic polynomial over $\mathbb{F}_q$ of degree $n$ asymptotically approaches the joint distribution of cycles of length $1$ through $k$ of a random permutation in $S_n$. I do not see how this result follows from the Chebotarev density theorem. 
