Duality of Graded Graphs Through Operads

Pairs of graded graphs, together with the Fomin property graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is one Young integer partitions, allowing us to connect number standard tableaux and numbers permutations. Here, we use operads, that algebraic devices abstracting notion composition objects, build pairs graphs. For this, first construct pair graphs where vertices syntax trees, elements free nonsymmetric operads. This dual new duality called $\phi$-diagonal similar ones introduced by Fomin. We also provide general way from wherein underlying posets analogous lattice. Some examples operads leading involving compositions, Motzkin paths, $m$-trees considered.