Year: 2015
Type: article
Abstract: Let |$i(n,k)$| be the proportion of permutations |$\pi \in \mathcal {S}_n$| having an invariant set of size |$k$|. In this note, we adapt arguments of the second author to prove that |$i(n,k) \asymp k^{-\delta } (1+\log k)^{-3/2}$| uniformly for |$1\leq k\leq n/2$|, where |$\delta = 1 - \frac {1 + \log \log 2}{\log 2}$|. As an application, we show that the proportion of |$\pi \in \mathcal {S}_n$| contained in a transitive subgroup not containing |$\mathcal {A}_n$| is at least |$n^{-\delta +o(1)}$| if |$n$| is even.
DOI: 10.1093/imrn/rnv371
Authors: Sean Eberhard, Kevin Ford, Ben Green
Institutions: University of Oxford, University of Illinois Urbana-Champaign
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1860069566