The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams
Abstract
John M. Campbell constructed a combinatorial Hopf algebra (CHA) ParSym on partition diagrams by lifting the CHA structure of NSym (the Hopf algebra of noncommutative symmetric functions) through an analogous approach. In this article, we define ParQSym, which is the graded dual of ParSym. Its CHA structure is defined in an explicit, combinatorial way, by analogy with that of the CHA QSym of quasisymmetric functions. And we give some subcoalgebra and Hopf subalgebras of ParQSym, some gradings and filtrations of ParSym and ParQSym, and some bases of ParSym and ParQSym by analogy with some distinguished bases of NSym and QSym.
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