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Generalized triangulations, pipe dreams, and simplicial spheres. (English. French summary) Zbl 1355.05078
Proceedings of the 23rd international conference on formal power series and algebraic combinatorics, FPSAC 2011, Reykjavik, Iceland, June 13–17, 2011. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 885-896 (2011).
Summary: We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between \(k\)-triangulations of the \(n\)-gon and \(k\)-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for \(k\)-triangulations with rotation to \(k\)-flagged tableaux with promotion.
For the entire collection see [Zbl 1239.05002].

MSC:
05B50 Polyominoes
05A15 Exact enumeration problems, generating functions
05E10 Combinatorial aspects of representation theory
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