  We present some Markovian approaches to prove universality results for some
functions on the symmetric group. Some of those statistics are already studied
in [Kammoun, 2018, 2020] but not the general case. We prove, in particular,
that the number of occurrences of a vincular patterns satisfies a CLT for
conjugation invariant random permutations with few cycles and we improve the
results already known for the longest increasing subsequence. The second
approach is a suggestion of a generalization to other random permutations and
other sets having a similar structure than the symmetric group.
