  In this paper we give a direct proof of the positivity conjecture for adapted
quantum cluster variables. Moreover, our process allows one to explicitly
compute formulas for all adapted cluster monomials and certain ordered products
of adapted cluster monomials. In particular, we describe all cluster monomials
in cluster algebras and quantum cluster algebras of rank 2. One may obtain
similar formulas for all finite type cluster monomials.
  The above results are achieved by computing explicit set-theoretic
decompositions of Grassmannians of subrepresentations in adapted valued quiver
representations into a disjoint union of products of standard vector space
Grassmannians. We actually prove a more general result which should be of
independent interest: we compute these decompositions for arbitrary flags of
subrepresentations in adapted valued quiver representations.
  This implies the existence of counting polynomials for the number of points
in these sets over different finite fields. Using this we extend the results of
\cite{rupel} to quantum cluster algebras $\Acal_q(Q,\bfd)$, where $q$ is an
indeterminate.
