  We continue our development of a new basis for the algebra of non-commutative
symmetric functions. This basis is analogous to the Schur basis for the algebra
of symmetric functions, and it shares many of its wonderful properties. For
instance, in this article we describe non-commutative versions of the
Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising
relation develops among non-commutative Littlewood-Richardson coefficients,
which has implications to the commutative case. Finally, we interpret these new
coefficients geometrically as the number of integer points inside a certain
polytope.

Combinatorics