  This paper proves some conjectures raised by N. Bergeron, N. Cartier, C.
Ceballos, and V. Pilaud about acyclic facets of subword complexes of type A
Coxeter groups. We use a representation of subword complex facets extending the
pipe dreams defined N. Bergeron and S. Billey and prove that the strongly
acyclic facets realize a lattice quotient of a weak order interval. We also
give a sufficient condition so that all acyclic facets are strongly acyclic.
