Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials

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 the language of k-flagged tableaux with promotion.