Permutads

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. A permutad is an algebra over the monad made of surjective maps between finite sets. It turns out that this notion is equivalent to the notion of “shuffle algebra” introduced previously 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.