  We are interested in two random matrix ensembles related to permutations: the
ensemble of permutation matrices following Ewens' distribution of a given
parameter $\theta >0$, and its modification where entries equal to $1$ in the
matrices are replaced by independent random variables uniformly distributed on
the unit circle. For the elements of each ensemble, we focus on the random
numbers of eigenvalues lying in some specified arcs of the unit circle. We show
that for a finite number of fixed arcs, the fluctuation of the numbers of
eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a
single arc, we extend this result to the case where the length goes to zero
sufficiently slowly when the size of the matrix goes to infinity. Finally, we
investigate the behaviour of the largest and smallest spacing between two
distinct consecutive eigenvalues.
