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This thesis comes within the scope of algebraic combinatorics and studiesproblems related to three orders on permutations: the two said weak orders(right and left) and the strong order or Bruhat order. The first part dealswith bases of multivariate polynomials. Most specifically, we study a productof Grothendieck polynomials and prove that it can interpreted as a sum over theBruhat order. We also present our implementation of Grothendieck polynomialsand other bases in Sage. In a second part, we study the Tamari order binarytrees. We obtain a new enumeration formula on the Tamari lattice and a newcombinatorial prove of Chapoton's functional equation of the generatingfunctions of Tamari intervals. We extend our results to the m-Tamari case andthus retrieve a formula given by Bousquet-M\'elou, Pr\'eville-Ratelle and Fusy.

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