A combinatorial basis for the free Lie algebra of the labelled rooted trees

TL;DR

This paper constructs an explicit basis for a sub operad related to labelled rooted trees, providing a new proof of its free twisted Lie algebra structure and revealing its operadic properties.

Contribution

It explicitly constructs a basis for the sub operad F within the pre-Lie operad T, clarifying its algebraic structure and operadic relationships.

Findings

F forms a sub nonsymmetric operad of T

Provides an explicit basis for F

Offers a new proof of Chapoton's results

Abstract

The pre-Lie operad can be realized as a space T of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra. That is, the S-module T is obtained as the plethysm of the S-module Lie with an S-module F. In the context of species, we construct an explicit basis of F. This allows us to give a new proof of Chapoton's results. Moreover it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie operad T.