3 problems

Let $\operatorname{Prim}'$ and $\operatorname{Cay}$ be the species of primitive objects and Cayley permutations, respectively. For a Cayley permutation $w=w_1\cdots w_n$, let … be…

Let $\mathscr{Y}=\mathscr{E}\circ\mathscr{D}^{-1}$ be the quotient series in the group $\mathsf{G}_{\operatorname{PreLie}}$, let $\mathscr{Y}_T$ be the coefficient of a rooted tree…

Let $\mathbf{g}$ be a positive graphical species, and let $\overrightarrow{\mathbf{A}}$ denote the graphical species of acyclic orientations. Write $\circ$ for the composition prod…