Cancelation free formula for the antipode of linearized Hopf monoid

Many combinatorial Hopf algebras $H$ in the literature are the functorial image of a linearized Hopf monoid $\mathbf{H}$. That is, $H=\mathcal{K}(\mathbf{H})$ or $H=\bar{\mathcal{K}}(\mathbf{H})$. Unlike the functor $\overline{\mathcal K}$,thefunctor${\mathcal K}$appliedto${\bf H}$ may not preserve the antipode of $\mathbf{H}$. In this case, one needs to consider the larger Hopf monoid $\mathbf{L}\times \mathbf{H}$ to get $H={\mathcal K} ({\bf H})=\overline{\mathcal K}({\bf L}\times{\bf H})$andstudytheantipodein${\bf L}\times{\bf H}$. One of the main results in this paper provides a cancelation free and multiplicity free formula for the antipode of ${\bf L}\times{\bf H}$\mathrm{.}Fromthisformulaweobtainanewantipodeformulafor$H$. We also explore the case when $\mathbf{H}$ is commutative and cocommutative. In this situation we get new antipode formulas that despite of not being cancelation free, can be used to obtain one for $\overline{\mathcal K}({\bf H})$ in some cases. We recover as well many of the well-known cancelation free formulas in the literature. One of our formulas for computing the antipode in $\mathbf{H}$ involves acyclic orientations of hypergraphs as the central tool. In this vein, we obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanley’s (-1)-color theorem. One of our examples introduces a {\it chromatic} polynomial for permutations which counts increasing sequences of the permutation satisfying a pattern. We also study the statistic obtained after evaluating such polynomial at $-1$. Finally, we sketch $q$ deformations and geometric interpretations of our results. This last part will appear in a sequel paper in joint work with J. Machacek.