The average order of a permutation
Let \(S_n\) be the symmetric group on \(n\) letters, and let \(N(\sigma)\) denote the group theoretic order of \(\sigma\in S_n\). \textit{P. Erdős} and \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 18, 309-320 (1967; Zbl 0235.20003)] proved that \(\log N(\sigma)\) has an asymptotically normal distribution with mean \(2^{-1}\log^2n\) and standard deviation \(3^{-1/2}\log^{3/2}n\). For other proofs, see \textit{M. R. Best} [Nederl. Akad. Wet., Proc., Ser. A 73, 385-402 (1970; Zbl 0208.03303)] and \textit{J. D. Bovey} [Bull. Lond. Math. Soc. 12, 41-46 (1980; Zbl 0436.20001)]. For a refined version, see \textit{J.-L. Nicolas} [Acta Math. Hung. 45, 69-84 (1985; Zbl 0574.10044)]. Let \(\mu_n\) denote the expectation of \(N(\sigma)\), i.e., \[ \mu_n= \sum_{\sigma\in S_n}N (\sigma)/n!. \] \textit{P. Erdős} and \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 19, 413-435 (1968; Zbl 0235.20004)] mentioned the estimation \(\log\mu_n= O((n/ \log n)^{1/2})\) without proof. \textit{E. Schmutz} [J. Number Theory 31, 260-271 (1989; Zbl 0675.10033)] proved that \(\log \mu_n<7.7(n/ \log n)^{1/2}\). \textit{W. M. Y. Goh} and \textit{E. Schmutz} [Bull. Lond. Math. Soc. 23, 34-42 (1991; Zbl 0735.11044)] proved that \(\log \mu_n \sim C(n/ \log n)^{1/2}\) for an explicit constant \(C\). In the paper under review the author proves that \[ \log\mu_n=C(n/ \log n)^{1/2} +O\bigl(n^{1/2} (\log\log n)/ \log n \bigr). \]
asymptotic formula for the expected order of a random permutation
scientific article; zbMATH DE number 1208673