The estimate of variances for random permutations

The purpose of this thesis was to evaluate the proportion between a variance of some dependent variables‘ sum and the sum of their variances. This problem was brought to number theory by J. Kubilius in 1985. Since then, new results were published but the exact estimate could not be found. The calculations in number theory had a huge impact in combinatorial problems. The coordinates of the cycle vector of a random permutation are dependent random variables. That is why the variance of a linear statistic is not the sum of its summands’ variances. E. Manstavičius has published some research addressing this problem. His recent paper, published jointly with J. Klimavičius, contains an optimal inequality for the uniformly sampled permutations of symmetric group. This thesis expands the latter by considering these permutations sampled according to the Ewens distribution with one parameter. This distribution was defined by W. J. Ewens in 1972 to investigate the genetical mutations. It is widely used in statistics and combinatorics