Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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.

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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con