# 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.

## Binary pattern of length greater than 14 are abelian-2-avoidable

Summary: Two words u and v are abelian equivalent if they are permutation of each other ("aabc" and "baca" are abelian equivalent). Let w be a word and P= P1...Pn (where the Pi are the letters of P) a pattern (a word over another alphabet), we say that w ...

## Cyclic sieving for reduced reflection factorizations of the Coxeter element

Given a factorization \(t_1\cdots t_n=c\) of some element \(c\) in a group, there are various natural cyclic operations we can apply on it; one of them is given by \(\Psi:(t_1,\cdots,t_n)\rightarrow (c\ t_n\ c^{-1},t_1,\cdots, t_{n-1})\). A common question is then to ...

## Convergence of uniform noncrossing partitions toward the Brownian triangulation

We give a short proof that a uniform noncrossing partition of the regular \(n\)-gon weakly converges toward Aldous's Brownian triangulation of the disk, in the sense of the Hausdorff topology. This result was first obtained by Curien and Kortchemski, using a more complicated encoding. Thanks to a result ...

Translations: fr