  We characterize the quotients among lattice path matroids (LPMs) in terms of
their diagrams. This characterization allows us to show that ordering LPMs by
quotients yields a graded poset, whose rank polynomial has the Narayana numbers
as coefficients.
  Furthermore, we study full lattice path flag matroids and show that --
contrary to arbitrary positroid flag matroids -- they correspond to points in
the nonnegative flag variety. At the basis of this result lies an
identification of certain intervals of the strong Bruhat order with lattice
path flag matroids.
  A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of
quotients of positroids. We use our results to prove this conjecture in the
case of LPMs.
