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Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition polynomials, which, when the indeterminates are reindexed, are comprised of the same partition monomials of those for the refined face partition polynomials of permutahedra and associahedra and the partition polynomials for noncrossing partitions (as indicated in this associated MO-Q).

These antipode polynomials, when reduced to graded polynomials in a single variable as are A133437 to A033282 for the associahedra and A049019 to A019538 for the permutohedra, apparently give the coarse face vectors of the simplicial noncrossing hypertree complexes of McCammond listed on pg. 15 of his paper "Noncrossing Hypertrees" (the polynomials of A102537).

The monomials of the refined face partition polynomials of the permutohedra and the associahedra label geometrically distinct faces of these convex polytopes, such as pentagons versus squares for the 2-D facets of the 3-D associahedron. See this MO-Q for a more explicit discussion for the permutahedra and this MO answer, for the associahedra.

What might the distinct monomials of the antipode partition polynomials designate geometrically?

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