An operational calculus for the Mould operad

TL;DR

This paper develops an operational calculus for the mould operad, simplifying its structure, discovering suboperads, proving a conjecture about non-crossing trees, and connecting it to noncommutative symmetric functions.

Contribution

It introduces an operational calculus for the mould operad, leading to simplifications, suboperad discoveries, and a proof of a conjecture relating non-crossing trees and the dendriform operad.

Findings

Proved a conjecture about the inverse image of non-crossing trees.

Discovered various suboperads within the mould operad.

Established a connection with noncommutative symmetric functions.

Abstract

The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture of the first author about the inverse image of non-crossing trees in the dendriform operad. Finally, we explain a connection with the formalism of noncommutative symmetric functions.