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