Abstract
We extend the definition of tridendriform bialgebra by introducing a parameter q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q-Gerstenhaber–Voronov algebras. We prove the equivalence between the categories of conilpotent q-tridendriform bialgebras and of q-Gerstenhaber–Voronov algebras. The space spanned by surjective maps between finite sets, as well as the space spanned by parking functions, have a natural structure of q-tridendriform bialgebra, denoted ST ( q ) and PQSym ( q ) ∗ , in such a way that ST ( q ) is a sub-tridendriform bialgebra of PQSym ( q ) ∗ . Finally we show that the bialgebra of M -permutations defined by T. Lam and P. Pylyavskyy comes from a q-tridendriform algebra which is a quotient of ST ( q ) .
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