Probability of generation by random permutations of given cycle type

Sean Eberhard, Daniele Garzoni

Suppose $\pi$ and ${\pi }^{\prime }$ are two random elements of $S_n$ with constrained cycle types such that $\pi$ has $x{n}^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise ${\pi }^{\prime }$ has $x' n^{1/…