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Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf monoid/algebra and a symmetric polynomial Hopf monoid/algebra, respectively (see Aguilar and Ardila, "Hopf monoids and generalized permutahedra"). Partition polynomials (A134264) constructed from the noncrossing partitions (NCPs) can also be used for compositional inversion. (All polytopes here and the NCPs are type A.)

Question: What role does this compositional inversion via the noncrossing partitions play in any Hopf monoids/algebras?

(Related discussions in "Hopf algebras and the logarithm of the S-transform in free probability" by Mastnak and Nica, "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" by Einziger, and "Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations" by Ebrahimi-Fard, Foissy, and Patras.)

(Edit Sept. 19, 2019)

It would probably suffice to unwrap this observation I found in "Faa di Bruno Hopf Algebras" by Figueroa, Gracia-Bondia, and Varilly:

Use of partitions with special properties may lead to other incidence algebras: for instance, if we restrict to noncrossing partitions, we obtain a cocommutative Hopf algebra, with the commutative group operation on characters essentially corresponding to Lagrange reversion of the Cauchy product of reverted series [cf. Stanley, Enumerative Comb., Vol. 2].

(But, it would be enlightening to understand the role of NCPs in other Hopf algebras as well.)

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  • $\begingroup$ Eqns. 6.27 on pg. 28 of Mastnak and Nica give an example of the NC polynomials $y_n$ that are the power series coefficients of the compositional inverse of $tH(-t)$ where $H(t)$ is the o.g.f. of the complete homogeneous symmetric polynomials. $\endgroup$
    – Tom Copeland
    Commented Sep 16, 2019 at 3:50
  • $\begingroup$ Also "Hopf algebras and dendriform structures arising from parking functions" by Novelli, Thibon arxiv.org/abs/math/0511200 $\endgroup$
    – Tom Copeland
    Commented Sep 18, 2019 at 20:41
  • $\begingroup$ And "Free cumulants, Schröder trees, and operads" by Josuat-Vergès, Menous, Novelli, and Thibon arxiv.org/abs/1604.04759 $\endgroup$
    – Tom Copeland
    Commented Sep 19, 2019 at 0:18
  • $\begingroup$ Also "The splitting process in free probability theory" by Kurusch Ebrahimi-Fard, Frederic Patras arxiv.org/abs/1502.02748 $\endgroup$
    – Tom Copeland
    Commented Dec 9, 2019 at 14:40

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