Invariable generation of the symmetric group

Abstract

We say that permutations $π_1,\dots, π_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $π'_1,\dots,π'_r$ of these permutations, $π'_1,\dots,π'_r$ generate $\mathcal{S}_n$. We show that if $π_1,π_2,π_3$ are chosen randomly from $\mathcal{S}_n$ then, with probability tending to 1 as $n \rightarrow \infty$, they do not invariably generate $\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate $\mathcal{S}_n$ with positive probability. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.