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 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/

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/