The # product in combinatorial Hopf algebras

TL;DR

This paper demonstrates that the # product of binary trees, initially introduced in a specific combinatorial context, can be generalized to the free associative algebra and extended to various classical combinatorial Hopf algebras.

Contribution

It reveals that the # product is definable at the algebraic level and extends beyond its original combinatorial setting to broader algebraic structures.

Findings

The # product is compatible with the free associative algebra.

The # product extends to most classical combinatorial Hopf algebras.

The algebraic formulation broadens the applicability of the # product.

Abstract

We show that the # product of binary trees introduced by Aval and Viennot [arXiv:0912.0798] is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.