Higher Homotopy Commutativity of H-Spaces and the Permuto-Associahedra

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an $A_n-space$. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected $A_p-space$ has the finitely generated mod p cohomology for a prime p and the multiplication of it is homotopy commutative of the p-th order, then it has the mod p homotopy type of a finite product of Eilenberg-Mac Lane spaces K(Z, 1)s, K(Z, 2)s and $K(Z/p^i, 1)s$ for i ≥ 1.