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A new hierarchy of operads over the linear spans of $delta$-cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of $delta$-cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite to the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, $m$-increasing trees, $c$-rectangular paths, and $m$-Dyck paths with operad structures. The operads on $c$-rectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads.
Using the combinatorial species setting, we propose two new operad structures on multigraphs and on pointed oriented multigraphs. The former can be considered as a canonical operad on multigraphs, directly generalizing the Kontsevich-Willwacher opera
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer partitions
We introduce a functorial construction $mathsf{C}$ which takes unitary magmas $mathcal{M}$ as input and produces operads. The obtained operads involve configurations of chords labeled by elements of $mathcal{M}$, called $mathcal{M}$-decorated cliques
We propose a new way of defining and studying operads on multigraphs and similar objects. For this purpose, we use the combinatorial species setting. We study in particular two operads obtained with our method. The former is a direct generalization o
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, opera