Combinatorics

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

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.

Invariant polynomial et théorème d’inversion sur le monoïde de Hopf des hypergraphes

-- Théo Karaboghossian (Labri, Université de Bordeaux)

La notion de monoïde de Hopf a été introduite par Aguiar et Mahajan et formalise de façon algébrique les notions de fusion et séparation d’objet combinatoires (concaténation de mots, restriction de graphes etc). Aguiar et Ardila ont montré que ce formalisme donne un cadre idéal pour définir des invariants ...

Type B extensions of Cauchy identity and Schur-positivity related to Chow’s quasisymmetric functions.

-- Alina Mayorova (Ecole Polytechnique)

The Cauchy identity is a fundamental formula in algebraic combinatorics that captures all the nice properties of the RSK correspondence. In particular, expanding both sides of the identity with Gessel's quasisymmetric functions allows to recover the descent preserving property, an essential tool to prove the Schur positivity of sets ...

La théorie équationnelle de l’ordre faible de Bruhat

-- Friedrich Wehrung (Université de Caen)

Ceci est un travail joint avec Luigi Santocanale. Il est connu que pour tout entier naturel n, le groupe symétrique d’ordre n peut être muni d’une structure de treillis, souvent appelé le permutoèdre sur n lettres P(n), qui est aussi l’ordre faible de Bruhat de type ...

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Translations: fr