Alternating strand diagrams and weakly separated sets

Alternating strand diagrams are generalizations of wiring diagrams, invented by Alex Postnikov to study total positivity in Grassmannians. A wiring diagram can be described by labeling the regions between the wires, which gives rise to a description in terms of the strongly separated collections of LeClerc and Zelevinsky. I will describe work with Oh and Postnikov showing that alternating strand diagrams likewise correspond to weakly separated collections. This leads to a proof of a conjecture of Scott and to a second proof of a conjecture of LeClerc and Zelevinsky (the first proof is in a recent preprint of Danilov, Karzanov and Koshevoy.)