On asymptotic expansions for the distribution of the number of cycles in a random permutation

A. N. Timashev

Diskretnaya Matematika, Tome 15 (2003) no. 3, pp. 117-127

Résumé

We obtain explicit formulas for the coefficients of asymptotic expansions in the domain of large deviations for the distributions of the number of cycles $\nu_n$ in a random permutation of degree $n$, that is, for the probability $\mathsf P\{\nu_n=N\}$ under the condition that $n,N\to\infty$ in such a way that $1\alpha_0\le\alpha=n/N\le\alpha_1\infty$, where $\alpha_0$, $\alpha_1$ are constants. These formulas express the coefficients in terms of cumulants of the random variable which has the distribution of the logarithmic series with specially chosen parameter. For the cumulants of the third and fourth orders we give the corresponding values. We discuss the accuracy of the obtained approximations. If $n,N\to\infty$ so that $0\gamma_0\le\gamma=N/\ln n\le\gamma_1\infty$, where $\gamma_0$, $\gamma_1$ are constants, we give asymptotic estimates of the probabilities $\mathsf P\{\nu_n=N\}$, $\mathsf P\{\nu_n\le N\}$, $\mathsf P\{\nu_n\ge N\}$ with the remainder terms of order $O((\ln n)^{-2})$ uniform in $\gamma\in[\gamma_0, \gamma_1]$. The corresponding estimate for the probability $\mathsf P\{\nu_n=N\}$ refines the previously known results for the case $N=\beta\ln n+o(\ln n)$, where $\beta$ is a positive constant.