We say that permutations invariably generate if, no matter how one chooses conjugates of these permutations, the permutations generate . We show that if , and are chosen randomly from , then, with probability tending to as , they do not invariably generate . By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate with probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.
"Invariable generation of the symmetric group." Duke Math. J. 166 (8) 1573 - 1590, 1 June 2017. https://doi.org/10.1215/00127094-0000007X