Ewens sampling and invariable generation

Combinatorics Seminar
Friday, January 26, 2018 - 15:00
1 hour (actually 50 minutes)
Skiles 005
Georgia Tech
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 Erdos, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula.