It is both free as an algebra and graded-cofree as a graded coalgebra, so is in some sense as far as possible from being either commutative or cocommutative.
The underlying free abelian group of the MPR algebra has a basis consisting of the disjoint union of the symmetric groups Sn for n = 0, 1, 2, .... , which can be thought of as permutations.
The coproduct of a permutation a on m points is given by Σa=b*c st(b) ⊗ st(c), where the sum is over the m + 1 ways to write a (considered as a sequence of m integers) as a concatenation of two sequences b and c, and st(b) is the standardization of b, where the elements of the sequence b are reduced to be a set of the form {1, 2, ..., n} while preserving their order.
The Hopf algebra of permutations relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions, (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram.
The duality between QSym and NSym is shown in the main diagonal of this diagram.