m-dendriform algebras

The Fuss-Catalan numbers are a generalization of the Catalan numbers. They enumerate a large class of objects and in particular m-Dyck paths and m+1-ary trees. Recently, F. Bergeron defined an analogue for generic m of the Tamari order on classical Dyck words. The author and J.-Y. Thibon showed that the combinatorial Hopf algebras related to these m-Tamari orders are defined thanks to the same monoid, the sylvester monoid, as in the m=1 case and that all related Hopf algebras also have m analogues. We present here the m-generalization of another construction on Catalan sets: the dendriform algebras. These algebras are presented in two different ways: first by relations between the m+1 operations, relations that are very similar to the classical relations; and then by explicit operations splitting the classical dendriform operations defined on words into new operations. We then investigate their dual and show they are Koszul.