Maximal fillings of Moon polyominoes, simplicial complexes, and Schubert polynomials (Q426771)

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 shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Moreover, for Ferrers shapes we 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 of length \(2(n-2k)\). Using this, we translate a conjectured cyclic sieving phenomenon for \(k\)-triangulations with rotation to the language of \(k\)-flagged tableaux with promotion.