The product of trees in the Loday-Ronco algebra through Catalan alternative tableaux

TL;DR

This paper introduces Catalan alternative tableaux to simplify the description of the product in the Loday-Ronco Hopf algebra of binary trees and proposes a new associative product on this space.

Contribution

It provides a novel combinatorial framework using tableaux to describe algebraic operations on binary trees, leading to a new associative product.

Findings

Simplified description of the Loday-Ronco product using tableaux

Introduction of a new associative binary tree product

Enhanced combinatorial understanding of binary tree algebra

Abstract

The aim of this note is to show how the introduction of certain tableaux, called Catalan alternative tableaux, provides a very simple and elegant description of the product in the Hopf algebra of binary trees defined by Loday and Ronco. Moreover, we use this description to introduce a new associative product on the space of binary trees.