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 \([n]=\{1,2,\ldots ,n\}\). 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.

我们考虑了在代数组合学和表示理论中具有有趣应用的三种类型的集合系统：集合的所谓强、弱和和弦分离子集的最大集合\([n]=\{1,2,\ ldots ,n\}\)。众所周知，这些系列具有很好的几何解释；即，它们是分别与在zonogon菱形拼接双射Ž（Ñ，2），合并的拼接上Ž（Ñ在3维zonotope，2），和细zonotopal拼接（或“cubillages”）Ž（n, 3)。我们在\(2^{[n]}\) 中描述了这三种类型的集合系统之间的相互关系，根据它们的几何模型工作。特别是，我们表征了可正确嵌入到固定 3 维立方体中的菱形和组合平铺的集合，并给出了将给定菱形或组合平铺扩展到立方体等的有效方法。