  Descent algebras of graded bialgebras were introduced by F. Patras as a
generalization of Solomon's descent algebras for Coxeter groups of type $A$,
i.e. symmetric groups. The universal enveloping algebra of the free Lie algebra
on a countable number of generators, its descent algebra and Solomon's descent
algebra, with its outer product, for symmetric groups are isomorphic to the
Hopf algebra of noncommutative symmetric functions, a free associative algebra
on a countable number of generators. In this paper we prove a similar result
for universal enveloping algebras of relatively free Sabinin algebras, which
makes them nonassociative analogues of the Solomon's descent algebra and of the
algebra of noncommutative symmetric functions. In the absolutely free case a
combinatorial setting is also presented by introducing a nonassociative lifting
of the Malvenuto-Reutenauer Hopf algebra of permutations and of the dual of the
Hopf algebra of noncommutative quasi-symmetric functions. We prove that this
lifting is a free nonassociative algebra as well as a cofree graded
coassociative coalgebra which also admits an associative inner product that
satisfies a Mackey type formula when restricted to its nonassociative Solomon's
descent algebra. Pseudo-compositions of sets instead of pseudo-compositions of
numbers is the natural language for the underlying combinatorics.
