Combinatoire alg\'ebrique li\'ee aux ordres sur les permutations

Abstract

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. The first part deals with bases of multivariate polynomials. Most specifically, we study a product of Grothendieck polynomials and prove that it can interpreted as a sum over the Bruhat order. We also present our implementation of Grothendieck polynomials and other bases in Sage. In a second part, we study the Tamari order binary trees. We obtain a new enumeration formula on the Tamari lattice and a new combinatorial prove of Chapoton's functional equation of the generating functions of Tamari intervals. We extend our results to the m-Tamari case and thus retrieve a formula given by Bousquet-M\'elou, Pr\'eville-Ratelle and Fusy.