Given the formal Taylor series, or e.g.f.,
$f(x) = 1 + \sum_{n \geq 1} m_n \; \frac{x^n}{n!}$,
the classical formal cumulants $c_n$ are generated from the formal moments $m_n$ via
$ \sum_{n \geq 1} c_n(m_1,...,m_n) \; \frac{x^n}{n!} = \ln[f(x)]$.
It's straightforward to show analytically that
$$\frac{\partial}{\partial m_k} \; c_n = \binom{n}{k} \; p_{n-k},$$
where $p_n(m_1,m_2,...,m_n)$ are the refined Euler characteristics partition polynomials, or signed refined face polynomials, of the permutohedra. (By convention the binomial coefficient vanishes for $k> n$.) The $p_n$ are designated $b_n$ in the examples in A133314 and are illustrated in the MO-Q "Relating face polytopes of permutohedra to integer partitions".
The first ten cumulant expansion polynomials $c_n(m_1, ...,m_n)$ are compiled by W. Lang in his pdf link in A127671.
Is there a combinatorial proof of this differential identity?
In parentheses in the table at the bottom of p. 2 of "The semi-classical approximation for modular operads" by Ezra Getzler, you will find the unsigned $c_n$ of A127671 and A263634. They are related to combinatorial necklaces in the text just above the table. The log and necklaces are related to the species of cyclic permutations (see "Species and Functors and Types, Oh My!" by Brent A. Yorgey).
The o.g.f. analog to the e.g.f. formulation above results from replacing the permutohedral polynomials of A133314, $p_n$, with those of A263633 mod signs (a refined Pascal matrix) and the cumulant expansion polynomials $c_n$ of A127671 with the Faber polynomials of A263916 mod signs. The corresponding diff id is eqn. 1.16 on p. 181 of "Differential calculus on the Faber polynomials" by Helene Airault and Abdlilah Bouali. This might be an easier formulation to approach combinatorically.
Example computation:
For $n=6$ and $k=2$,
$\frac{\partial}{\partial a_2} \; c_6 = \binom{6}{2}\; p_4 = 15 \; p_4.$
From A133314,
$15 \; p_4 = 15 \; (-a_4 + 8a_3a_1 + 6 a_2^2 - 36 a_2 a_1^2 + 24 a_1^4)$
$=-15a_4 + 120a_3a_1 + 90 a_2^2 - 540 a_2 a_1^2 + 360 a_1^4,$
and, from Lang's compilation in A127671, consistently
$\frac{\partial}{\partial a_2} \; c_6 =\frac{\partial}{\partial a_2} \;( 1 a_6 - 6 a_1a_5 - 15 a_2a_4 -10 a_3^2 +30 a_1^2 a_4 + 120 a_1a_2a_3 +30 a_2^3 - 120 a_1^3a_3 - 270 a_1^2a_2^2 + 360 a_1^4a_2 - 120 a_1^6)$
$ = - 15 a_2a_4 + 120 a_1a_2a_3 - 270 a_1^2a_2^2 + 360 a_1^4.$
An analogous, but more restricted, diff eq holds for the partition polynomials of A350499 that generate the free cumulants $c_n$ from the free moments in free probability theory:
$$\frac{\partial}{\partial m_1} \; c_n = n \; p_{n-k},$$
where here $p_n$ are the refined Euler characteristic partition polynomials of the associahedra. See the 1/20/22 edit in the MO-Q "Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory" for an example of this diff id. See section B of my answer to the MO-Q "Why is there a connection between enumerative geometry and nonlinear waves?" for an example $p_5=b_5$.
