  We investigate several Hopf algebras of diagrams related to Quantum Field
Theory of Partitions and whose product comes from the Hopf algebras WSym or
WQSym respectively built on integer set partitions and set compositions. Bases
of these algebras are indexed either by bipartite graphs (labelled or
unlabbeled) or by packed matrices (with integer or set coefficients).
Realizations on biword are exhibited, and it is shown how these algebras fit
into a commutative diagram. Hopf deformations and dendriform structures are
also considered for some algebras in the picture.
