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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 and generalizing usual configurations of chords. By considering combinatorial subfamilies of $mathcal{M}$-decorated cliques defined, for instance, by limiting the maximal number of crossing diagonals or the maximal degree of the vertices, we obtain suboperads and quotients of $mathsf{C} mathcal{M}$. This leads to a new hierarchy of operads containing, among others, operads on noncrossing configurations, Motzkin configurations, forests, dissections of polygons, and involutions. Besides, the construction $mathsf{C}$ leads to alternative definitions of the operads of simple and double multi-tildes, and of the gravity operad.
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 o
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
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
Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangians) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the Euclidean s