Enumerative and bijective combinatorics, permutation patterns, permutation sorting.

My research has focused on the bijective and enumerative aspects of finite combinatorial structures, falling within the Permutation Patterns discipline. Some topics I am interested in are:

Cayley permutations

Cayley permutations are words of positive integers containing at least one copy of each number between one and their maximum value. They naturally generalize permutations by allowing repeated entries, encode ordered set partitions, and are representatives of equivalence classes of integer words modulo pattern isomorphism. I want to explore their combinatorial and probabilistic aspects. I have recently employed combinatorial species to prove that the proportion of fixed-point-free Cayley permutations is asymptotic to 1/e, the same as permutations and endofunctions! What are some other interesting properties shared by these structures?

Fishburn structures

Ascent sequences were introduced in a very well known paper by Bousquet-Melou, Claesson, Dukes and Kitaev, where they play the role of  auxiliary objects that allow the authors to prove some nice bijective correspondences between various combinatorial structures: (2+2)-free posets, Stoimenow matchings and a set of pattern avoiding permutations, the so called Fishburn permutations. My goal is to develop a theory of transport for these two notions of pattern. Ideally, given a class of pattern avoiding Fishburn permutations, I aim to describe the corresponding set of ascent sequences in terms of pattern avoidance, and viceversa. I am also interested in several generalizations of this framework, including the d-Fishburn structures.

Permutation sorting

Given a system of devices (e.g. stacks, queues and some of their generalizations), I want to characterize and enumerate the set of permutations that can be sorted by this network. A permutation is sorted if the output is the identity string. The analysis of these devices can be often done in terms of permutation patterns. Another goal is to implement optimal sorting algorithms, i.e. procedures that are able to sort every sortable permutation, and study the related complexity. Sometimes I need to consider more specific algorithms (for example greedy-type procedures) that are easier to be studied, although less performing. I am interested in some dynamical properties of sorting devices too.

Genomic models

Genomic segments can be encoded as permutations, and genomic mutations as elementary operations that one can perform to transform a given permutation. Genomic mutations allow to define evolutionary distances on DNA strings as the minimum number of operations needed to transform a given string into another target string. This approach leads to combinatorial model that can be studied through the lens of pattern avoidance. For instance, patterns allow to characterize and enumerate the sets of permutation with fixed genomic distance with respect to a privileged string. The related algorithms for the computation of the genomic distance between two any strings and their complexity are also interesting.