Une op\'erade anticyclique sur les arbustes

TL;DR

This paper introduces a new combinatorial structure called shrubs, establishes an operad structure on them, and explores their relationship with Zinbiel operads and moulds, revealing compatibility with anticyclic structures.

Contribution

It defines shrubs as a new combinatorial object, constructs an operad on them, and demonstrates their inclusion in Zinbiel operads with compatible anticyclic structures.

Findings

Shrubs form a new combinatorial class related to rooted forests.

An operad structure on shrubs is constructed and analyzed.

The operad of shrubs is contained in the Zinbiel operad, compatible with anticyclic structures.

Abstract

We define new combinatorial objects, called shrubs, such that forests of rooted trees are shrubs. We then introduce a structure of operad on shrubs. We show that this operad is contained in the Zinbiel operad, by using the inclusion of Zinbiel in the operad of moulds. We also prove that this inclusion is compatible with the richer structure of anticyclic operad that exists on Zinbiel and on moulds.