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Massias, Jean-Pierre

Majoration explicite de l’ordre maximum d’un élément du groupe symétrique. (Explicit upper bound of the maximal order of an element of the symmetric group). (French) Zbl 0574.10043

Ann. Fac. Sci. Toulouse, V. Sér., Math. 6, 269-281 (1984).
Let g(n) denote the largest order of an element of the symmetric group of degree n. Landau showed that log g(n)\(\sim \sqrt{n \log n}\), as \(n\to \infty\). The author gives an explicit upper bound for \(f(n)=(\log g(n))/\sqrt{n \log n}\), namely \(f(n)<1.05314\) for all \(n\geq 1\), the maximal value being attained at \(n=1 319 766\).
Reviewer: A.Hildebrand

Cited in 16 Documents

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
11A25 Arithmetic functions; related numbers; inversion formulas
20B30 Symmetric groups
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\textit{J.-P. Massias}, Ann. Fac. Sci. Toulouse, Math. (5) 6, 269--281 (1984; Zbl 0574.10043)
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Online Encyclopedia of Integer Sequences:

Landau’s function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.
Numbers N in A002809 such that there is rho > 0 such that for all A > 0, A008475(A)-A008475(N) >= rho*log(A/N).
Related to a highly composite sequence (A002497).
Positions of running maxima of log(g(n))/sqrt(n*log(n)), where g(n) is Landau’s function A000793.

References:

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