We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a “Brownian separable permuton”.
"The Brownian limit of separable permutations." Ann. Probab. 46 (4) 2134 - 2189, July 2018. https://doi.org/10.1214/17-AOP1223