  We introduce, by adopting the point of view and the tools offered by the
theory of operads, a generalization on a nonnegative integer parameter $\gamma$
of diassociative algebras of Loday, called $\gamma$-pluriassociative algebras.
By Koszul duality of operads, we obtain a generalization of dendriform
algebras, called $\gamma$-polydendriform algebras. In the same manner as
dendriform algebras are suitable devices to split associative operations into
two parts, $\gamma$-polydendriform algebras seem adapted structures to split
associative operations into $2 \gamma$ operations so that some partial sums of
these operations are associative. We provide a complete study of the operads
governing our generalizations of the diassociative and dendriform operads.
Among other, we exhibit several presentations by generators and relations,
compute their Hilbert series, show that they are Koszul, and construct free
objects in the corresponding categories. We also provide consistent
generalizations on a nonnegative integer of the duplicial, triassociative and
tridendriform operads, and of some operads of the operadic butterfly.
