On maximal weakly separated set-systems

Abstract

For a permutation $\omega \in S_n$, Leclerc and Zelevinsky \cite{LZ} introduced a concept of $\omega$-{\em chamber weakly separated collection} of subsets of $\{1,2,...,n\}$ and conjectured that all inclusion-wise maximal collections of this sort have the same cardinality $\ell (\omega )+n+1$, where $\ell (\omega )$ is the length of $\omega$. We answer affirmatively this conjecture and present a generalization and additional results.

36 pages. This is the final version for J. Algebr. Comb

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