# Combinatorics

The main focus of this activity is the interrelation between algebraic structure and algorithms. We plan to work on the following subjects:

- Algebraic structures (Combinatorial Hopf Algebras, Operads, Monoids, ...) related to algorithms;
- Enumerative combinatorics and symbolic dynamic.
- Object oriented software design for modeling mathematics and development of SageMath;

More precisely, the research project takes place in effective algebraic combinatorics, at the interface of enumerative combinatorics and analysis of algorithms on one hand and symbolic and algebraic computation on the other hand. The objective is twofold: firstly, thanks to vast generalization of the notion of generating series, we hope to give a theoretical framework allowing to study the fine behavior of various algorithms. Reciprocally, the study of those very same algorithms gives a new mean to discover algebraic identities. Those identities have many applications in mathematics, in particular in representation theory but also in physics (mainly statistical physics).

The research relies deeply on computer experimentation and contains as a consequence an important software development part within the Sage-Combinat software project. However, the required level of sophistication, flexibility, and breath of computational tools is reaching a point where large scale collaborative development is critical. The design and collaborative development of such a software is raising research-grade computer science challenges around the modelling of mathematics, the management of large hierarchy of (object oriented) classes, etc.

Those very specific questions also raise more general combinatorial questions. We therefore plan to work on enumerative combinatorics and cellular automaton, in particular on trees. This activity is conducted with close collaborators in France, Germany, North America, and India.

## Ranking aggregation: graph-based methods and use in bioinformatics

summary: The problem of ranking aggregation is the following: we have a set of elements and a set of rankings of these elements as input, and we want a single ranking as output, which best reflects the set of rankings taken as input. The applications are manifold, especially in bioinformatics ...

## Algèbres tridendriformes, arbres de Schröder et algèbre de Hopf

Les concepts d’algèbres dendriformes, respectivement tridendriformes décrivent l’action de certains éléments du groupe symétrique appelés les battages et respectivement les battages contractants sur l’ensemble des mots dont les lettres sont des éléments d’un alphabet, respectivement d'un monoïde. Un lien entre les algèbres dendriformes et tridendriformes ...

## Skipless chain decompositions and improved poset saturation bounds

summary: We show that given m disjoint chains in the Boolean lattice, we can create m disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). By using this result we are able to answer ...

Translations: fr