4 problems

Let $\textbf{\textsc{pre}}^\natural$ denote the Hopf monoid arising from a Hopf class; equivalently, consider every Hopf monoid arising from a Hopf class. Multiplicity-freeness con…

Let $\textbf{\textsc{bc}}$, $\textbf{\textsc{omat}}^{\operatorname{\mathsf{gale}}}$, $\textbf{\textsc{qe}}$, and $\textbf{\textsc{qi}}$ denote the corresponding classes of prefix-p…

Let the universal Hopf class be denoted by $\textbf{\textsc{pre}}$. Multiplicity-freeness conjecture. The antipode for the Hopf monoid of the universal Hopf class…

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…