Cayley graph diameters for fixed cycle types are eventually quasipolynomial

Source abstract

Given a cycle type, the corresponding conjugacy class of $S_n$ generates either $S_n$ or $A_n$ for sufficiently large $n$. We prove that the sequence of diameters of Cayley graphs is eventually polynomial on residue classes for any fixed cycle type. This result is also extended to finite unions of conjugacy classes.