The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams

John M. Campbell constructed a Combinatorial Hopf Algebra (CHA) ParSym on partition diagrams by lifting the CHA structure of NSym, which is the Hopf algebra of noncommutative symmetric functions. In this paper, we construct the graded dual ParQSym of ParSym. Similar to the construction of CHA QSym of quasisymmetric functions, the CHA structure of ParQSym is described in an explicit, combinatorial way. In addition, we present some subcoalgebras and Hopf subalgebras of ParQSym, gradings and filtrations of ParSym and ParQSym, and certain bases of ParSym and ParQSym by analogy with the distinguished bases of NSym and QSym.