  Motivated by work of Buch on set-valued tableaux in relation to the K-theory
of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf
algebras can be thought of as K-theoretic analogues of the by now classical
``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric
functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf
algebra of permutations. In addition, we develop a theory of set-valued
P-partitions and study three new families of symmetric functions which are
weight generating functions of reverse plane partitions, weak set-valued
tableaux and valued-set tableaux.
