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The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamariorder defined on objects counted by Fuss-Catalan numbers, such as m-Dyck pathsor (m+1)-ary trees. On another hand, the Tamari order is related to the productin the Loday-Ronco Hopf algebra of planar binary trees. We introduce newcombinatorial Hopf algebras based on (m+1)-ary trees, whose structure isdescribed by the m-Tamari lattices.  In the same way as planar binary trees can be interpreted as sylvesterclasses of permutations, we obtain (m+1)-ary trees as sylvester classes of whatwe call m-permutations. These objects are no longer in bijection withdecreasing (m+1)-ary trees, and a finer congruence, called metasylvester,allows us to build Hopf algebras based on these decreasing trees. At theopposite, a coarser congruence, called hyposylvester, leads to Hopf algebras ofgraded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functionsand quasi-symmetric functions in a natural way. Finally, the algebras of packedwords and parking functions also admit such m-analogues, and we present theirsubalgebras and quotients induced by the various congruences.

Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions