  The immaculate basis of the non-commutative symmetric functions was recently
introduced by the first and third author to lift certain structures in the
symmetric functions to the dual Hopf algebras of the non-commutative and
quasi-symmetric functions. It was shown that immaculate basis satisfies a
positive, multiplicity free right Pieri rule. It was conjectured that the left
Pieri rule may contain signs but that it would be multiplicity free. Similarly,
it was also conjectured that the dual quasi-symmetric basis would also satisfy
a signed multiplicity free Pieri rule. We prove these two conjectures here.
