# ALGEBRAIC COMBINATORICS

Random generation with cycle type restrictions
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 1-25.

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $\pi ,{\pi }^{\prime }\in {S}_{n}$, we prove that $\pi$ and a random conjugate of ${\pi }^{\prime }$ are likely to generate at least ${A}_{n}$ provided only that $\pi$ and ${\pi }^{\prime }$ have not too many fixed points and not too many $2$-cycles. As an application, we investigate the following question: For which positive integers $m$ should we expect two random elements of order $m$ to generate ${A}_{n}$? Among other things, we give a positive answer for any $m$ having any divisor $d$ in the range $3\le d\le o\left({n}^{1/2}\right)$.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.149
Classification: 20B30,  20P05
Keywords: symmetric group, random generation.
@article{ALCO_2021__4_1_1_0,
author = {Eberhard, Sean and Garzoni, Daniele},
title = {Random generation with cycle type restrictions},
journal = {Algebraic Combinatorics},
pages = {1--25},
publisher = {MathOA foundation},
volume = {4},
number = {1},
year = {2021},
doi = {10.5802/alco.149},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.149/}
}
Eberhard, Sean; Garzoni, Daniele. Random generation with cycle type restrictions. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 1-25. doi : 10.5802/alco.149. https://alco.centre-mersenne.org/articles/10.5802/alco.149/

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