A Sequent Calculus for a Semi-Associative Law

We introduce a sequent calculus with a simple restriction of Lambek\u27s product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. Elsewhere, we have also used the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a natural fragment of lambda calculus, consisting of the beta-normal planar lambda terms with no closed proper subterms

Number of right ideals and a qq-analogue of indecomposable permutations

We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field $\mathbb F\_q$ is equal to $(q-1)^{n+1} q^{\frac{(n+1)(n-2)}{2}}\sum\_\theta q^{inv(\theta)}$, where the sum is over all indecomposable permutations in $S\_{n+1}$ and where $inv(\theta)$stands for the number of inversions of $\theta$.Comment: submitte

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

A Symmetry Preserving Algorithm for Matrix Scaling

International audienceWe present an iterative algorithm which asymptotically scales the $\infty$-norm of each row and each column of a matrix to one. This scaling algorithm preserves symmetry of the original matrix and shows fast linear convergence with an asymptotic rate of $1/2$. We discuss extensions of the algorithm to the one-norm, and by inference to other norms. For the 1-norm case, we show again that convergence is linear, with the rate dependent on the spectrum of the scaled matrix. We demonstrate experimentally that the scaling algorithm improves the conditioning of the matrix and that it helps direct solvers by reducing the need for pivoting. In particular, for symmetric matrices the theoretical and experimental results highlight the potential of the proposed algorithm over existing alternatives.Nous décrivons un algorithme itératif qui, asymptotiquement, met une matrice à l'échelle de telle sorte que chaque ligne et chaque colonne est de taille 1 dans la norme infini. Cet algorithme préserve la symétrie. De plus, il converge assez rapidement avec un taux asymptotique de 1/2. Nous discutons la généralisation de l'algorithme à la norme 1 et, par inférence, à d'autres normes. Pour le cas de la norme 1, nous établissons que l'algorithme converge avec un taux linéaire. Nous démontrons expérimentalement que notre algorithme améliore le conditionnement de la matrice et qu'il aide les méthodes directes de résolution en réduisant le pivotage. Particulièrement pour des matrices symétriques, nos résultats théoriques et expérimentaux mettent en valeur l'intérêt de notre algorithme par rapport aux algorithmes existants

Cooperative tasking for multi-agent systems

Ph.DDOCTOR OF PHILOSOPH