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 $bf H$. 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 Stanleyu0027s (-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.