Patterns in random permutations avoiding the pattern 132

We consider a random permutation drawn from the set of 132-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after scaling by $n^{\lambda(\sigma)/2}$ where $\lambda(\sigma)$ is the length of $\sigma$ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.Comment: 32 page

The Brownian limit of separable permutations

We study random uniform 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".Comment: 45 pages, 14 figures, incorporating referee's suggestion

Area of Catalan Paths on a Checkerboard

Abstract. It is known that the area of all Catalan paths of length n is equal to 4n − `2n+1 ´ , which coincides n with the number of inversions of all 321-avoiding permutations of length n + 1. In this paper, a bijection between the two sets is established. Meanwhile, a number of interesting bijective results that pave the way to the required bijection are presented. Résumé. Le fait que la somme des surfaces des chemins Catalan de longueur n est égale à 4n − `2n+1 ´ , ce n qui est aussi le nombre d’inversions dans toutes les permutations de longueur n + 1 qui évitent le motif 321, est bien connu. Nous présentons dans cet article une bijection entre ces deux ensembles. Pour ce faire, nous établissons plusieurs résultats bijectifs intermédiaires intéressants. 1