Geodesic
On the number of permutations admitting an \(m\)th root
The electronic journal of combinatorics, Tome 9 (2002)
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Let $m$ be a positive integer, and $p_n(m)$ the proportion of permutations of the symmetric group $S_n$ that admit an $m$-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula $$p_n(m)\, \sim \, {{\pi _m}\over {n^{1-\varphi (m)/m}}},\;\; n\to \infty ,$$ where $\varphi$ is the Euler function and $\pi _m$ an explicit constant.
DOI : 10.37236/1620
Classification : 05A15, 05A16, 68W40
Mots-clés : number of permutations, exponential generating function, asymptotic formula 
@article{10_37236_1620,
     author = {Nicolas Pouyanne},
     title = {On the number of permutations admitting an \(m\)th root},
     journal = {The electronic journal of combinatorics},
     year = {2002},
     volume = {9},
     doi = {10.37236/1620},
     zbl = {0990.05003},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1620/}
}
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Nicolas Pouyanne. On the number of permutations admitting an \(m\)th root. The electronic journal of combinatorics, Tome 9 (2002). doi: 10.37236/1620

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