John M. Campbell constructed a combinatorial Hopf algebra (CHA) \text{ ParSym} on partition diagrams by lifting the CHA structure of \text{ NSym} (the Hopf algebra of noncommutative symmetric functions) through an analogous approach. In this article, we define \text{ ParQSym}, which is the graded dual of \text{ ParSym}. Its CHA structure is defined in an explicit, combinatorial way, by analogy with that of the CHA \text{ QSym} of quasisymmetric functions. And we give some subcoalgebra and Hopf subalgebras of \text{ ParQSym}, some gradings and filtrations of \text{ ParSym} and \text{ ParQSym}, and some bases of \text{ ParSym} and \text{ ParQSym} by analogy with some distinguished bases of \text{ NSym} and \text{ QSym}.
Categories: math.RA