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- W4300623105 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 $leftlceil ( 1- alpha log 2 )^{-1} rightrceil$ 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 ErdH{o}s, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula." @default.
- W4300623105 created "2022-10-03" @default.
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- W4300623105 date "2016-10-13" @default.
- W4300623105 modified "2023-09-24" @default.
- W4300623105 title "Ewens sampling and invariable generation" @default.
- W4300623105 doi "https://doi.org/10.48550/arxiv.1610.04212" @default.
- W4300623105 hasPublicationYear "2016" @default.
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