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

For the ordered set [n] of n elements, we consider the class B-n of bases B of tropical Plucker 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 B-n, yielding the affirmative answer to one conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex Delta(m)(n) = {S subset of [n]: [S] = m}. (C) 2009 Elsevier Inc. All rights reserved.