Permutads

TL;DR

This paper introduces permutads, a new algebraic structure related to shuffle algebras and permutohedra, providing a noncommutative analogue of nonsymmetric operads and connecting to existing concepts like shuffle operads.

Contribution

It defines permutads, establishes their equivalence to shuffle algebras, and relates them to permutohedra, expanding the framework of operad theory.

Findings

Permutads are a new algebraic structure controlling specific word compositions.

Permutads are equivalent to shuffle algebras and closely related to shuffle operads.

The role of the associahedron is played by the permutohedron in permutad theory.

Abstract

We unravel the algebraic structure which controls the various ways of computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a new type of operads, that we call permutads. It turns out that this notion is equivalent to the notion of "shuffle algebra" introduced by the second author. It is also very close to the notion of "shuffle operad" introduced by V. Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the notion of nonsymmetric operads. We show that the role of the associahedron in the theory of operads is played by the permutohedron in the theory of permutads.