  Motivated by a question in Schubert calculus, we study the interplay of
quasisymmetric polynomials with the divided symmetrization operator, which was
introduced by Postnikov in the context of volume polynomials of permutahedra.
Divided symmetrization is a linear form which acts on the space of polynomials
in $n$ indeterminates of degree $n-1$. We first show that divided
symmetrization applied to a quasisymmetric polynomial in $m$ indeterminates can
be easily determined. Several examples with a strong combinatorial flavor are
given. Then, we prove that the divided symmetrization of any polynomial can be
naturally computed with respect to a direct sum decomposition due to
Aval-Bergeron-Bergeron involving the ideal generated by positive degree
quasisymmetric polynomials in $n$ indeterminates.
