  Using the combinatorics of $\alpha$-unimodal sets, we establish two new
results in the theory of quasisymmetric functions. First, we obtain the
expansion of the fundamental basis into quasisymmetric power sums. Secondly, we
prove that generating functions of reverse $P$-partitions expand positively
into quasisymmetric power sums. Consequently any nonnegative linear combination
of such functions is $p$-positive whenever it is symmetric. As an application
we derive positivity results for chromatic quasisymmetric functions,
unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials
and the more general $B$-polynomials, matroid quasisymmetric functions, and
certain Eulerian quasisymmetric functions, thus reproving and improving on
numerous results in the literature.
