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.
Non-intersecting paths and the determinant of the distance matrix of a tree
summary:
We present a combinatorial proof of the Graham–Pollak formula for the determinant of the
distance matrix of a tree, via sign-reversing involutions and the Lindström–Gessel–Viennot Lemma.
This is joint work with Emmanuel Briand, Luis Esquivias-Quintero, Álvaro Gutierrez, and Adrián Lillo.
Lattice structures of Gog and Magog triangles
summary:
Gog and Magog triangles are simple combinatorial objects which are equienumerated.
Howewer, the problem of finding an explicit bijection between these has been an
open problem since the 80’s. These are related to other interesting objects such
as alternating sign matrices, plane partitions or aztec diamond tillings.
All ...
Une nouvelle description des treillis m-cambriens
summary:
Les treillis cambriens, introduits par N. Reading en 2006, sont une généralisation du treillis de Tamari, à tout choix d'élément de Coxeter, dans tout groupe de Coxeter fini. Le treillis de Tamari correspond au "type A linéaire". Ces ordres partiels admettent plusieurs descriptions, non trivialement équivalentes. Celles-ci donnent ...Alternating and nondeterministic plane-walking automata
summary: Plane-walking automata were introduced by Salo & Törma to recognise languages of two-dimensional infinite words (subshifts), the counterpart of 4-way finite automata for two-dimensional finite words. We extend the model to allow for nondeterminism and alternation of quantifiers. We prove that the recognised subshifts form a strict subclass of sofic ...
À la lumière de quelques propriétés essentielles
Cet exposé consistera en quatre parties, chacune commençant par une introduction à un domaine de recherche et terminant par quelques contributions.
Nous débuterons par expliquer comment la conjecture polynomiale de Hales-Jewett à densité unifie plusieurs des généralisations du théorème de van der Waerden, ainsi que comment cette généralisation commune présente ...
Translations: fr