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Effective approach of the tridendriform Schroeder tree algebra

Pierre Catoire, Jean Fromentin

Abstract

We introduce a primitive computation problem in the free tridendriform algebra generated by one element which is a Hopf algebra based on Schroeder trees. We know a complex way to generate all of them. To understand it clearer, we want to implement this method on a computer. However, we need to create some tools to implement Schroeder trees and the multiplications over this algebra to be able to compute the primitive elements. We also checked numerically that they are all primitive elements. In this paper, we detail how we made the problem mathematically understandable for a computer and how we implement it.

Effective approach of the tridendriform Schroeder tree algebra

Abstract

We introduce a primitive computation problem in the free tridendriform algebra generated by one element which is a Hopf algebra based on Schroeder trees. We know a complex way to generate all of them. To understand it clearer, we want to implement this method on a computer. However, we need to create some tools to implement Schroeder trees and the multiplications over this algebra to be able to compute the primitive elements. We also checked numerically that they are all primitive elements. In this paper, we detail how we made the problem mathematically understandable for a computer and how we implement it.

Paper Structure

This paper contains 35 sections, 9 theorems, 51 equations, 2 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Aim and structure of the paper
  3. Table of notations
  4. Acknowledgments
  5. About the original problem
  6. Schroeder trees and comb representations
  7. Our objective
  8. Our point of view of Schroeder trees and their operations
  9. Encoding Schroeder trees
  10. Levelled Schroeder Tree
  11. Our point of view on Schroeder trees
  12. Our choice of section
  13. The encoding
  14. The tridendriform structure on $\mathop{\mathrm{Im}}\nolimits(\mathop{\mathrm{Code}}\nolimits)$
  15. Preliminary definitions
  16. ...and 20 more sections

Key Result

Lemma 7

Let $k,l\geq 2$. Then, the following sets are in bijections: For other cases, we have:

Figures (2)

  • Figure 1: Diagram of the generation of primitives by induction
  • Figure 2: Scheme to enumerate quasi-shuffles

Theorems & Definitions (60)

  • Definition 1
  • Example 2
  • Definition 4
  • Definition 5
  • Definition 6: Quasi-shuffle
  • Lemma 7
  • Definition 8
  • Example 9
  • Theorem 10: Catoire_2023
  • Theorem 11
  • ...and 50 more