# How often are Galois groups equal to $S_n$?

Let $$\mathbb{Z}[x]_n$$ be the set of polynomials in $$\mathbb{Z}[x]$$ of degree at most $$n$$.

Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$

of $$\mathbb{Z}[x]_n$$ by finite sets, i.e., a sequence of nested, finite sets $$A_k$$ such that $$\bigcup_k A_k = \mathbb{Z}[x]_n$$. (One possible sensible option is to take $$A_k$$ to be the set of polynomials whose coefficients are all in $$\{-k, \ldots, k\}$$.)

For each $$A_k$$ and each subgroup $$H \leq S_n$$ (up to isomorphism), we can ask for the proportion $$p_k(H)$$ of polynomials in $$A_k$$ whose Galois group has isomorphism type $$H$$.

What can be said about the limiting behavior of $$p_k(H)$$ as $$k \to \infty$$ that doesn't depend too severely on the filtration $$(A_k)$$? In particular, is the Galois group of a randomly selected polynomial of degree $$< n$$ almost surely $$S_n$$, or more precisely, is $$\lim_{k \to \infty} p_k(S_n) = 1?$$

If not, what is this probability? If it is $$0$$ or $$1$$, what can be said about the limiting behavior of $$p_k(S_n)$$?

What about these questions for other subgroups of $$S_n$$?

S. D. Cohen proved that the proportion of polynomials in $$\mathbb{Z}[x]_n$$ and coefficients all at most $$k$$ is absolute value for which the Galois group is not $$S_n$$ is $$\ll \frac{\log k}{\sqrt k},$$ so $$\lim_{k \to \infty} p_k(S_n) = 1 ,$$ and we have a lower bound on convergence.
Remark None of the sources seem, however, to give lower bounds for $$p_k(H)$$ for proper subgroups $$H < S_n$$, i.e., a description of the asymptotics of how often particular proper subgroups occur as Galois groups of polynomials in $$\mathbb{Z}[x]_n$$.