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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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The first few partition polynomials (ParPs) of the set $[A^{(2)}]$ that comprise Einziger's antipode are

$A^{(2)}_0 = 1$

$A^{(2)}_1 = - u_1 $

$A^{(2)}_2 = - u_2 + 3 u_1^2 $

$A^{(2)}_3 = - u_3 + 8 u_1 u_2 - 12 u_1^3 $

$A^{(2)}_4 = −u_4 + 10 u_1 u_3 + 5 u_2^2 − 55 u_1^2 u_2 + 55 u_1^4 .$

I fairly recently discovered that Isaac Newton presented these ParPs as the compositional inversion coefficients for odd o.g.f.s in a letter to Oldenburg in 1676 (one of two secretaries of the Royal Society and a sometimes mischievous clearinghouse for correspondence between the Continent and Britain--such was suspect by the Crown, always on the lookout for intrigue).

So, these are the ParPs for the compositional inversion

$$(O^{(2)}(z))^{(-1)} = z + A^{(2)}_1(u_1)z^3 + A^{(2)}_2(u_1,u_2) z^5 + A^{(2)}_3(u_1,u_2,u_3) z^7 +\cdots$$

of an odd o.g.f

$$O^{(2)}(z)= z + u_1 z^3 + u_2 z^5 + u_2 z^5 +\cdots,$$

and, as such inherit the diverse combinatorial interpretations of the monomials of the set $[A]$ of ParPs of A133437 / A111785, the refined signed face, or the refined Euler characteristic, polynomials of the associahedra (see, e.g., the MO-Q "Guises of the associahedra" and the refs in the OEIS entries), the coefficients for compositional inversion of generic formal power series, or ordinary generating functions.

Refinements of these polynomials with other combinatorial characterizations are presented by Novelli and Thibon in "Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra" (in fact, they ref Einziger and A102537, a natural reduction of these polynomials, in which Einziger had previously been referenced).

The larger algebra in which $[A^{(2)}]$ is embedded is presented in MO_Q1, MO-Q2 , and MO-Q3.

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