Algebraic combinatorics on orders of permutations
type: Dataset
description: This thesis comes within the scope of algebraic combinatorics and studies problems related to three orders on permutations: the two said weak orders (right and left) and the strong order or Bruhat order.We first look at the action of the symmetric group on multivariate polynomials. By using the emph{divided differences} operators, one can obtain some generalisations of the Schur function and form bases of non symmetric multivariate polynomials. This construction is similar to the one of Schur functions and also allows for geometric interpretations. We study more specifically the Grothendieck polynomials which were introduced by Lascoux and Schützenberger. Lascoux proved that a product of these polynomials can be interpreted in terms of a product of divided differences. By developing this product, we reobtain a result of Lenart and Postnikov and also prove that it can be interpreted as a sum over an interval of the Bruhat order. We also present our implementation of multivariate polynomials in Sage. This program allows for formal computation on different bases and also implements many changes of bases. It is based on the action of the divided differences operators. The bases include Schubert polynomials, Grothendieck polynomials and Key polynomials. In a second part, we study the emph{Tamari lattice} on binary trees. This lattice can be obtained as a quotient of the weak order. Each tree is associated with the interval of its linear extensions. We introduce a new object called, emph{interval-posets} of Tamari and show that they are in bijection with the intervals of the Tamari lattice. Using these objects, we give the recursive formula counting the number of elements smaller than or equal to a given tree. We also give a new proof that the generating function of the intervals of the Tamari lattice satisfies some functional equation given by Chapoton. Our final contributions deals with the m-Tamari lattices. This family of lattices is a generalization of the classical …
landing page: https://pastel.hal.science/tel-00952773
identifier: NNT: 2013PEST1093
keyword: algebraic-combinatorics
keyword: algebre-de-hopf
keyword: combinatoire-algebrique
keyword: differences-divisees
keyword: divided-differences
keyword: grothendieck-polynomials
keyword: hopf-algebra
keyword: infoeu-reposemanticsdoctoralthesis
keyword: infoinfo-ohcomputer-science-csother-csoh
keyword: orders-of-the-symmetric-group
keyword: ordres-du-groupe-symetrique
keyword: polynomes-de-grothendieck
keyword: tamari-lattice
keyword: theses
keyword: treillis-de-tamari
issued: 2026-05-06T05:25:10.755069
modified: 2026-05-06T05:25:10.755074
language: fr
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Algebraic combinatorics on orders of permutations
type: Distribution
format: HTML
access URL: https://pastel.hal.science/tel-00952773
issued: 2026-05-06T05:25:10.785753
modified: 2026-05-06T05:25:10.741560