  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. 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 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.
