Asymptotic behavior of some statistics in Ewens random permutations.

*(English)*Zbl 1314.05016Ewans random model for permutations is a way to choose a permutation out of all of size \(N\). It depends on one input-parameter and gives, as a special case, the uniform distribution model. In this work several parameters (statistics) of such a randomly chosen permutation are analyzed as \(N\) tends to infinity. These include, among others, the number of cycles, exceedences (a parameter coming from statistical physics), adjacencies, and, more generally, the number of occurrences of any dashed pattern.

All the results are derived from one main theorem. To apply this theorem, you need to properly define the random variables corresponding to the desired parameter. This results in “small” joint cumulants, which fits very well the intuition one can have about such problems. The resulting analysis of the parameters then gives convergence to a distribution. The main theorem is general enough to be applied to other problems as well. Its proof uses the ideas of the method of moments.

Although some of the results on the above described parameters were already known in the literature, this work provides one proof for all of them and extends these results further.

All the results are derived from one main theorem. To apply this theorem, you need to properly define the random variables corresponding to the desired parameter. This results in “small” joint cumulants, which fits very well the intuition one can have about such problems. The resulting analysis of the parameters then gives convergence to a distribution. The main theorem is general enough to be applied to other problems as well. Its proof uses the ideas of the method of moments.

Although some of the results on the above described parameters were already known in the literature, this work provides one proof for all of them and extends these results further.

Reviewer: Daniel Krenn (Graz)