  It is known from the work of Baik, Deift, and Johansson [1999] that we have
Tracy-Widom fluctuations for the longest increasing subsequence of uniform
permutations. In this paper, we prove that this result holds also in the case
of the Ewens distribution and more generally for a class of random permutation
with distribution invariant under conjugation. Moreover, we obtain the
convergence of the first components of the associated Young tableaux to the
Airy Ensemble as well as the global convergence to the
Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that
the limiting descent process of a large class of random permutations is
stationary, one-dependent and determinantal.
