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

TL;DR

This thesis explores algebraic combinatorics related to permutation orders, including polynomial bases, Bruhat order, and Tamari lattices, providing new formulas, interpretations, and implementations.

Contribution

It introduces new enumeration formulas, combinatorial proofs, and computational implementations for permutation orders and related algebraic structures.

Findings

Product of Grothendieck polynomials interpreted as sum over Bruhat order

New enumeration formula for Tamari lattice

Extension of results to m-Tamari case

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.