  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.
