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We extend the definition of tridendriform bialgebra by introducing a weightq. The subspace of primitive elements of a q-tridendriform bialgebra isequipped with an associative product and a natural structure of brace algebra,related by a distributive law. This data is called q-Gerstenhaber-Voronovalgebras. We prove the equivalence between the categories of connectedq-tridendriform bialgebras and of q-Gerstenhaber-Voronov algebras. The spacespanned by surjective maps, as well as the space spanned by parking functions,have natural structures of q-tridendriform bialgebras, denoted ST(q) andPQSym(q)*, in such a way that ST(q) is a sub-tridendriform bialgebra ofPQSym(q)*. Finally we show that the bialgebra of M-permutations defined by T.Lam and P. Pylyavskyy may be endowed with a natural structure ofq-tridendriform algebra which is a quotient of ST(q).

Tridendriform structure on combinatorial Hopf algebras