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We are concerned with permutations taken uniformly at random from the symmetric group. Firstly, we study the probability of a permutation missing short cycles. Secondly, the result is employed to establish a formula for total variation distance between the process of multiplicities of cycle lengths in a random permutation and a process of independent Poisson random variables. We apply an analytic...
We examine the asymptotic distribution laws of integer-valued linear statistics defined via the multiplicities of lengths of cycles comprising a random permutation taken with equal probability. We establish necessary and sufficient conditions for the weak convergence to the Poisson limit law. The approach can be applied to investigate other logarithmic structures.
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