  We explore the asymptotic distributions of sequences of integer-valued
additive functions defined on the symmetric group endowed with the Ewens
probability measure as the order of the group increases.
  Applying the method of factorial moments, we establish necessary and
sufficient conditions for the weak convergence of distributions to discrete
laws.
  More attention is paid to the Poisson limit distribution. The particular case
of the number-of-cycles with restricted lengths function is analyzed in more
detail. The results can be applied to statistics defined on random permutation
matrices.
