Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

For the ordered set [n] of n elements, we consider the class Bn of bases B of tropical Plücker functions on ²[n] such that B can be obtained by a series of so-called weak flips (mutations) from the basis formed by the intervals in [n]. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on an n-zonogon. Based on the generalized tiling representation, we then prove that each weakly separated set-system in ²[n] having maximum possible size belongs to Bn, yielding the affirmative answer to one conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex .