On interrelations between strongly, weakly and chord separated set-systems (a geometric approach)

We consider three types of set-systems that have interesting applications in algebraic combinatorics and representation theory: maximal collections the so-called strongly, weakly, chord separated subsets a set $$[n]=\{1,2,\ldots ,n\}$$ . These are known to admit nice geometric interpretations; namely, they are, respectively, bijection with rhombus tilings on zonogon Z(n, 2), combined fine zonotopal (or “cubillages”) 3-dimensional zonotope 3). describe interrelations between these $$2^{[n]}$$ , working terms their models. In particular, we characterize sets properly embeddable fixed cubillage give efficient methods extending given or tiling cubillage, etc.