Jean Fromentin

Exploring the tree of numerical semigroups

In this paper we describe an algorithm visiting all numerical semigroups up to a given genus using a well suited representation. The interest of this algorithm is that it fits particularly well the architecture of modern computers allowing very large optimizations: we obtain the number of numerical semigroups of genus g 67 and we confirm the Wilf conjecture for g 60.

Investigating monte-carlo methods on the weak schur problem

Nested Monte-Carlo Search (NMC) and Nested Rollout Policy Adaptation (NRPA) are Monte-Carlo tree search algorithms that have proved their efficiency at solving one-player game problems, such as morpion solitaire or sudoku 16x16, showing that these heuristics could potentially be applied to constraint problems. In the field of Ramsey theory, the weak Schur numberWS(k) is the largest integer n for which their exists a partition into k subsets of the integers [1,n] such that there is no x<y<z all in the same subset with x+y=z. Several studies have tackled the search for better lower bounds for the Weak Schur numbers WS(k), k≥4. In this paper we investigate this problem using NMC and NRPA, and obtain a new lower bound for WS(6), namely WS(6)≥582.

THE WELL-ORDERING OF DUAL BRAID MONOID

We describe the restriction of the Dehornoy ordering of braids to the dual braid monoids introduced by Birman, Ko and Lee: we give an inductive characterization of the ordering of the dual braid monoids and compute the corresponding ordinal type. The proof consists in introducing a new ordering on the dual braid monoid using the rotating normal form of arXiv:math.GR/0811.3902, and then proving that this new ordering coincides with the standard ordering of braids.

Every braid admits a short sigma-definite expression

A result by Dehornoy (1992) says that every nontrivial braid admits a σ -definite expression, defined as a braid word in which the generator σi with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a σ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.

Near-misses in Wilf’s conjecture

Let

A divisibility result in combinatorics of generalized braids

S \subseteq \mathbb N

A well-ordering of dual braid monoids

S⊆N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let