Counting Polynomials via Galois Actions on Root Subsets

This paper studies the number of monic integer polynomials $f$ of height at most $H$ whose Galois group, endowed with the action on the roots, is isomorphic to a prescribed permutation group $(G,\Omega)$. New upper bounds are obtained for several families of groups: transitive subgroups of the wreath product $S_m\wr S_r$ in the primitive action; $k$-homogeneous subgroups of $S_m$ in the action on $k$-subsets of $\{1,\ldots,m\}$; $k$-transitive subgroups of $S_m$ in the action on $k$-tuples of distinct elements of $\{1,\ldots,m\}$. Almost all finite groups in their regular permutation representation are also treated.