In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an
A
n
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
A_p
-space has the finitely generated mod
p
p
cohomology for a prime
p
p
and the multiplication of it is homotopy commutative of the
p
p
-th order, then it has the mod
p
p
homotopy type of a finite product of Eilenberg-Mac Lane spaces
K
(
Z
,
1
)
K(\mathbb {Z},1)
s,
K
(
Z
,
2
)
K(\mathbb {Z},2)
s and
K
(
Z
/
p
i
,
1
)
K(\mathbb {Z}/p^i,1)
s for
i
≥
1
i\ge 1
.