Ewens sampling and invariable generation

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 α-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 α/k. The special case α =1 corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed. For strong α-logarithmic measures, and almost every α, we show that precisely ⌈ ( 1- αlog 2 )^-1⌉ 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.