Dendriform algebras form a category of algebras recently introduced by Loday.A dendriform algebra is a vector space endowed with two nonassociative binaryoperations satisfying some relations. Any dendriform algebra is an algebra overthe dendriform operad, the Koszul dual of the diassociative operad. Weintroduce here, by adopting the point of view and the tools offered by thetheory of operads, a generalization on a nonnegative integer parameter $\gamma$of dendriform algebras, called $\gamma$-polydendriform algebras, so that$1$-polydendriform algebras are dendriform algebras. For that, we consider theoperads obtained as the Koszul duals of the $\gamma$-pluriassociative operadsintroduced by the author in a previous work. In the same manner as dendriformalgebras are suitable devices to split associative operations into two parts,$\gamma$-polydendriform algebras seem adapted structures to split associativeoperations into $2\gamma$ operation so that some partial sums of theseoperations are associative. We provide a complete study of the$\gamma$-polydendriform operads, the underlying operads of the category of$\gamma$-polydendriform algebras. We exhibit several presentations bygenerators and relations, compute their Hilbert series, and construct freeobjects in the corresponding categories. We also provide consistentgeneralizations on a nonnegative integer parameter of the duplicial,triassociative and tridendriform operads, and of some operads of the operadicbutterfly.
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