We investigate algebraic structures that can be placed on vertices of the
multiplihedra, a family of polytopes originating in the study of higher
categories and homotopy theory. Most compelling among these are two distinct
structures of a Hopf module over the Loday-Ronco Hopf algebra.

We use Hopf algebras to prove a version of the Littlewood-Richardson formula
for skew Schur functions, which implies a conjecture of Assaf and McNamara. We
also establish a similar skew Littlewood-Richardson formula for Schur P- and
Q-functions.

The multiplihedra {M_n} form a family of polytopes originating in the study
of higher categories and homotopy theory. While the multiplihedra may be
unfamiliar to the algebraic combinatorics community, it is nestled between two
families of polytopes that certainly are not: the permutahedra {S_n} and
associahedra {Y_n}. The maps between these families reveal several new Hopf
structures on tree-like objects nestled between the Malvenuto-Reutenauer (MR)
Hopf algebra of permutations and the Loday-Ronco (LR) Hopf algebra of planar
binary trees.