Ewens sampling and invariable generation

Abstract

We study the number of random permutations needed to invariably generate the symmetric group, $S_n$, when the distribution of cycle counts has the strong $\alpha$-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of $k$-cycles relates to a conditioned Poisson random variable with mean $\alpha /k$. The special case $\alpha =1$ corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed. For strong $\alpha$-logarithmic measures, and almost every $\alpha$, we show that precisely $\lceil (1-\alpha \log 2{)}^{-1}\rceil$ permutations are needed to invariably generate $S_n$. A corollary is that for many other probability measures on $S_n$ no bounded number of permutations will invariably generate $S_n$ with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula.

34 pages, 1 figure