  We show a precise formula, in the form of a monomial, for certain families of
parabolic Kazhdan-Lusztig polynomials of the symmetric group.
  The proof stems from results of Lapid-Minguez on irreducibility of products
in the Bernstein-Zelevinski ring. By quantizing those results into a statement
on quantum groups and their canonical bases, we obtain identities of
coefficients of certain transition matrices that relate Kazhdan-Lusztig
polynomials to their parabolic analogues.
  This affirms some basic cases of conjectures raised recently by Lapid.
