There are several ways of finding the power or Taylor series for the compositional inverse of a function $f(x)$ with $f(0)=0\;$ given its series expansion, e.g., by using the classic Lagrange inversion formula (see OEIS A145271 and cross references therein).
A more roundabout way is through binomial Sheffer polynomials (BSPs) and matrix inversion.
First, generate the BSP sequence $p_n(t)$ from
$$\exp[t \cdot f(x)] = \exp[x \cdot p.(t)] = \sum_{n \ge 0} p_n(t) \cdot \frac{x^n}{n!}.$$
Note that the Taylor series coefficients of $f(x)$ (multiplied by $t$) then can be generated as the cumulants produced from the moments $p_n(t)$ through the cumulant expansion OEIS A127671 since the natural logarithm of the LHS gives $t \cdot f(x)$.
Next, form the lower triangular matrix of coefficients of these row polynomials out to a suitable order, and then generate its matrix inverse, producing the BSPs $ \bar{p}_n(t)$ associated to the inverse function $f^{(-1)}(x).$ (See the footnote.)
The Sheffer formalism tells us that
$$\exp[x \cdot \bar{p}.(t)] = \exp[t \cdot f^{-1}(x)],$$
so, as mentioned above for $f(x)$, the cumulants of the sequence $\bar{p}_n(t)$ using the cumulant expansion formula A127671 are the Taylor series coefficients multiplied by $t$ of $f^{(-1)}(x).$
Each of these operations--the exponentiation, matrix inversion, cumulant formation--has a combinatorial and/or geometric interpretation.
My question: Is there a coherent overarching geometric/combinatorial interpretation of this sequence of operations of generating the Taylor series of an inverse function?
The pair of lower triangular matrices with row polynomials associated to the BSPs for $f(x)$ and its inverse are a matrix inverse pair. This follows from the results of umbral composition of the pair of BSP sequences:
$$ \exp[x \cdot p.(\bar{p}.(t)] = \exp[f(x) \cdot \bar{p}.(t)] $$
$$= \exp[t \cdot f^{(-1)}(f(x))] = \exp[tx];$$
therefore, the umbral composition yields
$$p_n(\bar{p}.(t))= t^n = \bar{p}_n(p.(t)),$$
which is equivalent to the corresponding matrices of coefficients of these BSPs being matrix inverses.
