  We study quadri-algebras and dual quadri-algebras. We describe the free
quadri-algebra on one generator as a subobject of the Hopf algebra of
permutations FQSym, proving a conjecture due to Aguiar and Loday, using that
the operad of quadri-algebras can be obtained from the operad of dendriform
algebras by both black and white Manin products. We also give a combinatorial
description of free dual quadri-algebras. A notion of quadri-bialgebra is also
introduced, with applications to the Hopf algebras FQSym and WQSym.

Combinatorics