Many combinatorial Hopf algebras $H$ in the literature are the functorial image of a linearized Hopf monoid $\bf H$. That is, $H={ \mathcal K} ({ \bf H})$ or $H=\overline{ \mathcal K} ({ \bf H})$. Unlike the functor $\overline{ \mathcal K}$, the functor ${ \mathcal K}$ applied to ${ \bf H}$ may not preserve the antipode of ${ \bf H}$. In this case, one needs to consider the larger Hopf monoid ${ \bf L}\times{ \bf H}$ to get $H={ \mathcal K} ({ \bf H})=\overline{ \mathcal K}({ \bf L}\times{ \bf H})$ and study the antipode in ${ \bf L}\times{ \bf H}$. One of the main results in this paper provides a cancelation free and multiplicity free formula for the antipode of ${ \bf L}\times{ \bf H}$. From this formula we obtain a new antipode formula for $H$. We also explore the case when ${ \bf H}$ 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 ${ \bf H}$ 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.
Categories: math.CO