We explore the probability ν(n, r) that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r, where 1 ≤ r ≤ n and n → ∞. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.
On the class of labelled combinatorial structures called assemblies we define complex-valued
multiplicative functions and examine their asymptotic mean values. The problem reduces
to the investigation of quotients of the Taylor coefficients of exponential generating series
having Euler products. Our approach, originating in probabilistic number theory, requires
information on the generating functions only in the convergence disc and rather weak
smoothness on the circumference. The results could be applied to studying the asymptotic
value distribution of decomposable mappings defined on assemblies.
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