  In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated
collections of subsets of the ordered $n$-element set $[n]$ (using this notion
to give a combinatorial characterization for quasi-commuting minors of a
quantum matrix). They conjectured the purity of certain natural domains
$D\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the
hyper-simplex $\{X\subseteq[n]\colon |X|=m\}$ for $m$ fixed), where $D$ is
called pure if all maximal weakly separated collections in $D$ have the same
cardinality. These conjectures have been answered affirmatively.
  In this paper, generalizing those earlier results, we reveal wider classes of
pure domains in $2^{[n]}$. This is obtained as a consequence of our study of a
novel geometric--combinatorial model for weakly separated set-systems,
so-called \emph{combined (polygonal) tilings} on a zonogon, which yields a new
insight in the area.
