  Using the theory of noncommutative symmetric functions, we introduce the
higher order peak algebras, a sequence of graded Hopf algebras which contain
the descent algebra and the usual peak algebra as initial cases (N = 1 and N =
2). We compute their Hilbert series, introduce and study several combinatorial
bases, and establish various algebraic identities related to the multisection
of formal power series with noncommutative coefficients.
