# Jacopo ## Phase transition for almost square permutations

A record in a permutation is an entry which is either bigger or smaller than the entries either before or after it (there are four types of records). Entries which are not records are called internal points. We explore scaling limits (called permuton limits) of uniform permutations in the classes Sq(n,k) of almost square permutations of size n+k with exactly k internal points.We …

## Almost square permutations are typically square (with Enrica Duchi and Erik Slivken)

A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal points. Permutations without internal points have been studied underthe name of square …

## The feasible region for consecutive patterns of permutations is a cycle polytope (with Raul Penaguiao)

We denote the proportion of consecutive patterns in a permutation as We consider  the consecutive pattern limiting sets, called the feasible region for consecutive patterns, defined for every   We are able to obtain a full description of the feasible region as the cycle polytope of a specific graph, called the overlap graph . Definition: …

## Programming 2019

This is the webpage for the course of Programming. Teacher: Asieh Parsania Lectures: Online at the following link. Exercises: Fr 15.00 – 17.00, Room: Y27H52

## Combinatorics of words 2019

This is the webpage for the course of Combinatorics of words. Teacher: Mathilde Bouvel Lectures: Th 15.00 – 17.00, Room: Y27H28 UZH-Webpage of the course I’ll weekly post here the lecture notes of the course: Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 …

## An approach through random walks to local limits of permutations encoded by generating trees

We develop a general technique to study local limits of random permutations in families enumerated through generating trees. We propose an algorithm to sample uniform permutations in such families using random walks and we recover the local behavior of permutations studying local properties of the corresponding random walks. The method applies to families of permutations …

## Square permutations are typically rectangular

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record.  First we describe the global behavior by showing these permutations have a permuton limit which can be described by a random rectangle.  We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions.  Second, we …

## A decorated tree approach to random permutations in substitution-closed classes (with Mathilde Bouvel, Valentin Féray, and Benedikt Stufler)

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local …

## Square permutations are typically rectangular (with Erik Slivken). Annals of Applied Probability, to appear.

We can think of the records of a permutation (i.e., left-to-right or right-to-left maxima or minima) as the external points of a permutation. The points of a permutation that do not correspond to records are called internal points.  Square permutations are permutations with no internal points. Here an example and a counterexample: The left permutation …

## What is… a permuton?

How does a large random permutation behave? We will try to answer this question for different classical models of random permutations, such as uniform permutations, pattern-avoiding permutations, Mallows permutations and many others. An appropriate framework to describe the asymptotic behaviour of these pemutations is to use a quite recent notion of scaling limits for permutations, …