A Combinatorial Hopf Algebra on Partition Diagrams

We introduce a Combinatorial Hopf Algebra (CHA) with bases indexed by the partition diagrams indexing the bases for partition algebras. By analogy with the operation $H_{\alpha} H_{\beta} = H_{\alpha \cdot \beta}$ for the complete homogeneous basis of the CHA $\textsf{NSym}$ given by concatenating compositions $\alpha$ and $\beta$, we mimic this multiplication rule by setting $\textsf{H}_{\pi} \textsf{H}_{\rho} = \textsf{H}_{\pi \otimes \rho}$ for partition diagrams $\pi$ and $\rho$ and for the horizontal concatenation $\pi \otimes \rho$ of $\pi$ and $\rho$. This gives rise to a free, graded algebra $\textsf{ParSym}$, which we endow with a CHA structure by lifting the CHA structure of $\textsf{NSym}$ using an analogue, for partition diagrams, of near-concatenations of integer compositions. Unlike the Hopf algebra $\textsf{NCSym}$ on set partitions, the new CHA $\textsf{ParSym}$ projects onto $\textsf{NSym}$ in natural way via a ``forgetful'' morphism analogous to the projection of $\textsf{NSym}$ onto its commutative counterpart $\textsf{Sym}$. We prove, using the Boolean transform for the sequence $(B_{2n} : n \in \mathbb{N})$ of even-indexed Bell numbers, an analogue of Comtet's generating function for the sequence counting irreducible permutations, yielding a formula for the number of generators in each degree for $\textsf{ParSym}$, and we prove, using a sign-reversing involution, an evaluation for the antipode for $\textsf{ParSym}$. An advantage of our CHA being defined on partition diagrams in full generality, in contrast to a previously defined Hopf algebra on uniform block permutations, is given by how the coproduct operation we have defined for $\textsf{ParSym}$ is such that the usual diagram subalgebras of partition algebras naturally give rise to Hopf subalgebras of $\textsf{ParSym}$ by restricting the indexing sets of the graded components to diagrams of a specified form.