The set $[N]$ of partition polynomials (ParPs) $N_n(u_1,...,u_n)$ of OEIS A134264 have numerous manifestations and applications in diverse areas of mathematics and physics (see the OEIS entry and this MO-Q on some).
Because of their intimate relation with compositional inversion of formal power series / ordinary generating functions (o.g.f.s)
$O(z;c.) = z \; ( 1 + c_1 z + c_2 z^2 + \cdots) = z / h(z;u.),$
one way of generating the polynomials is via the formalism of the classic Lagrange inversion formula (LIF):
with
$h(z;u.) = 1 + u_1 z + u_2 x^2 + \cdots = \frac{z}{O(z;c.)},$
$\frac{\partial^{n+1}_{\omega=0}}{(n+1)!} \; O^{(-1)}(\omega;c.) = N_n(u_1,...,u_n) = \frac{\partial_{z=0}^n}{n!} \frac{[1 + u_1z + u_2 z^2 +\cdots]^{n+1}}{n+1}.$
An o.g.f. for the first few is
$N(x) = 1 + \sum_{n \geq 1} N_n(u_1,...,u_n) x^n$
$= 1 + u_1 x + (u_2 + u_1^2)x^2 + (u_3 + 3 u_1 u_2 + u_1^3)x^3 + (u_4 + 4 u_1 u_3+ 2 u_2^2 + 6 u_1^2 u_2 + u_1^4) x^4 + (u_5 + 5 u_1 u_4 + 5 u_2 u_3 + 10 u_1^2 u_3 + 10 u_1 u_2^2 + 10 u_1^3 u_2 + u_1^5) x^5 + \cdots.$
The coefficient of the monomial $u_1^{e_1}u_2^{e_2}\cdots u_n^{e_n}$, with $Se = \sum_{k=1}^n e_k$, is
$$\frac{1}{n+1} \frac{(n+1)!}{(n+1-Se)!} \frac{1}{e_1!e_2!\cdots e_n!}= \frac{1}{n+1} \binom{n+1}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
Using the generalized Leibniz / Newton formula for products and the LIF, it is straightforward to prove the derivational identity
$(\dagger)$
$$\partial_{u_1} \; N_n(u_1,...,u_n) = n \; N_{n-1}(u_1,...,u_{n-1}).$$
Since $N_0=1$, this is sufficient to show that $[N]$ is an Appell Sheffer polynomial sequence and, therefore, the binomial convolution identity
$(\dagger\dagger)$
$$N_n(u_1+t,u_2,..,u_n) = (N.(u_1,u_2,...,u_n) + t)^n = \sum_{k=0}^n \binom{n}{k} N_k(u_1,u_2,...,u_k) t^{n-k} $$ holds.
(This follows from $\partial_{u_1} N(x) = (x\partial_xx)\; N(x)$ and is explicit in the e.g.f. for any Appell sequence.)
Reduced with $u_1 =0$ and $u_k = 1$ otherwise and $t =1$, ($\dagger \dagger$) becomes
$$c_n = (1 + m.)^n$$
where $c_n$ are the Catalan numbers of A000108, and $(m.)^n = m_n$, the Riordan numbers / Motzkin summands of A005043, with refinement A249548. Perhaps arguments for this reduced identity in "Catalan, Motzkin, and Riordan numbers" by Bernhart can be generalized. (An addition to the varied references in A005043 is "Semiorders and Riordan Numbers" by Barry Balof and Jacob Menashe.)
Question: What are some combinatorial proofs of the derivational $(\dagger)$ and binomial $(\dagger\dagger)$ identities for these partition polynomials for compositional inversion?
(The lit on these partition polynomials is broad and deep with geometric combinatorial models encompassing noncrossing partitions, Dyck lattice paths, parking functions, and trees. Somewhere there must be a comb. proof of these identities.)
