Higher homotopy commutativity of -spaces and the permuto-associahedra

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an -space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected -space has the finitely generated mod cohomology for a prime and the multiplication of it is homotopy commutative of the -th order, then it has the mod homotopy type of a finite product of Eilenberg-Mac Lane spaces s, s and s for .