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The central limit theorem for the number of different prime factors of a natural number (Erd\"{o}s-Kac, 1939) and that for the number of cycles of a random permutation in the symmetric group (Goncharov, 1942) were established almost at the same time. Nowadays, being a bit ahead probabilistic number theory supplies ideas to carry out parallel theories on permutations, all mappings of a finite set into itself or general classes of decomposable structures like assemblies, weighted multisets \textit{et cetera} (see \cite{ABT}). Going along this path, we explore various statistics of random permutations $\sigma$ taken at random from the symmetric group $\S$. Some of the recent results will be discussed in the talk. For illustration, we present here an analog of the weighted Tur\'{a}n-Kubilius inequality.

Let $k_j(\sigma)\geq 0$, $1\leq j\leq n$, be the number of cycles in the canonical decomposition of $\sigma$ into a product, $w(\sigma)=k_1(\sigma)+\cdots+k_n(\sigma)$ be the number of all cycles, $\Theta^{(n)}:=\theta(\theta+1)\cdots(\theta+n-1)$ where $\theta>0$ is a constant. The Ewens probability measure in $\S$ is defined by $\nu_n(\{\sigma\})=(\Theta^{(n)})^{-1} \theta^{w(\sigma)}$. By definition, a real additive function has an expression $h(\sigma)= h_1\big(k_1(\sigma)\big)+\cdots+h_n\big(k_n(\sigma)\big),$ where $h_j(s)\in \R$ and $h_j(0)\equiv 0$. For its variance with respect to $\nu_n$, we have \[{\mathbf V}_n h(\sigma)\leq C(\theta)\sum_{jk\leq n}\Big({\theta\over j}\Big)^k{h_j(k)^2\over k!} \bigg(1-{jk\over n+1}\bigg)^{\theta-1}. \] The ideas of Bir\'{o} and Szamuely \cite{BiSza-AMH96} lead to further inequalities with multiplicative weights.

References
[1] R. Arratia, A.D. Barbour and S. Tavare. Logarithmic Combinatorial Structures: a Probabilistic Approach. EMS Publishing House Zurich (2003).
[2] A. Biro and T. Szamuely, A Turan-Kubilius inequality with multiplicative weights. Acta Math. Hung., 70(1-2), 39-56 (1996).

randon permutation, cycle decomposition, Ewens probability, additive function