Three particularly important reps of the exponential formula (cf. MO-Q) are the

  1. refined Lah polynomials (OEIS A130561): Exp[o.g.f.] = Exp[formal power series]$\; =\exp[\frac{1}{(1-a.x)}]$, umbrally with $(a.)^n = a_n$ and $(a.)^0 = a_0=0$
  2. refined Stirling polynomials of the second kind, or Bell polynomials (A0036040): Exp[e.g.f.] = Exp[formal Taylor series] $\; = \exp[e^{b.x}]$ with $b_0=0$
  3. refined Stirling polynomials of the first kind, or cycle index polynomials (CIPs) of the symmetric groups (A0036039): Exp[l.g.f.] = Exp[logarthmic rep]$ = \exp[-\ln(1-c.x)]$.

The Bell partition polynomials are characterized by the Faa di Bruno Hopf algebra (cf. MO-Q for refs on Hopf algebras in general) with the antipode given by A134685. The antipodes for the refined Lah polynomials and CIPs are A133932 and A133437.

In the HA formalism, the explicit, graded formulas for the antipode are usually derived through an algebraic recursion, as naturally befits an algebraic characterization. For our three sets of partition polynomials, the antipode is the Lagrange inversion formula, which can also be expressed at each order $x^n$ as an iteration of the Lie derivative $g(x) \frac{d}{dx}= \frac{d}{df(x)}$, or infinitesimal generator (cf. MO-Q), where $f(x)$ is expressed as a formal o.g.f, e.g.f., or l.g.f., with $f(x)=0$.

Question 1: How can this differential relation for the antipode be explicitly expressed in a Hopf algebra formalism?

In addition, the refined Lah polynomials $L_n(a_1,..,a_n)$ are integer normalized versions of the elementary Schur polynomials, and, therfore, have the generalized lowering operators

$$\frac{d}{d(a_m)} L_n(a_1,..,a_n) = \frac{n!}{(n-m)!} L_{n-m}(a_1,..,a_{n-m}).$$

Question 2: How can this be expressed in the Hopf algebra formalism?

There are raising operators that can also be expressed as differential operators, so my question really boils down to how differental op characterizations of the partition polynomials translate into the HA formalism (and is certainly related to an expression of formal group laws $f^{-1}(f(x)+f(y))$ in the HA formalism).


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