Ewens sampling and invariable generation

TL;DR

This paper determines the number of random permutations needed to invariably generate the symmetric group under Ewens sampling measures, revealing a precise formula depending on the parameter lpha, and generalizes classical theorems in permutation group theory.

Contribution

It provides an exact formula for the number of permutations required for invariable generation under Ewens measures and extends classical theorems to these probabilistic settings.

Findings

For lpha=1, four permutations suffice, matching previous results.

For strong lpha-logarithmic measures, the required permutations depend on lpha via a specific formula.

Many measures on S_n do not allow a bounded number of permutations to invariably generate the group.

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 $\left\lceil ( 1- \alpha \log 2 )^{-1} \right\rceil$ 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 Erd\H{o}s, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula.