Define the associahedra partition polynomial $$ \begin{split} A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\ & \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) z^n \end{split} $$ with $$ \begin{split} A_n(u_1,...,u_n) &= \sum_\text{partitions of $n$} M(e_1,e_2,...,e_n) u_1^{e_1}u_2^{e_3}\ldots u_n^{e_n}\\ &= \sum_\text{partitions of $n$} (-1)^{Se} \frac{1}{n+1} \frac{(n+Se)!}{n!e_1!e_2!\ldots e_n!} u_1^{e_1}u_2^{e_3}\ldots u_n^{e_n} \end{split} $$ for which the sum of the exponents $e_n$ is $Se = e_1 + e_2 + \cdots + e_n$ and a partition of $n$ satisfies $n = 1 \cdot e_1 + 2 \cdot e_2 + \cdots + n \cdot e_n$ where $n$ and $e_n$ are natural numbers, $(0,1,2,...)$, $z$ is a complex or real variable, and $(u_1,u_2,...)$ is an infinite set of mutually commutative indeterminates that commute with $z$ as well.
In analysis, $z \; A(z)$ is the formal compositional inverse of the formal power series / o.g.f. $O(z) = z(1+ \sum_{n \geq 1} u_1 z^n)$. In geometry / topology, the associahedra partition polynomials $A_n(u_1,...,u_n)$ are essentially the refined Euler characteristic polynomials of the convex associahedra polytopes, i.e., associahedrons. Unsigned, they are the refined face polynomials of the associahedra, flagging and enumerating the geometrically distinct faces of the associahedra. (See OEIS A133437, especially links therein to several of my MO-Q&As for more info.)
A 'sampling theorem' applies to these partition polynomials (ParPs) such that given a 'periodically sampled' infinite subset of monomials of the ParPs, the $A_n$ can be completely generated.
Explicitly, with the sampled subset
$$A_n^{(m)}(\hat{u}_1,...,\hat{u}_n) = A_{nm}(0,...,0,u_m,0,...,0,u_{2m},0,...,0,u_{3m},0,...,0,u_{nm})$$
where $\hat{u}_n = u_{nm}$, then the lowering operation, a combination of multiplicative and compositional inversion,
$$ \frac{x}{(xA^{(m)}(x))^{(-1)}} = A^{(m-1)}(x) $$
can be used iteratively to generate $A(x)$.
There is a raising operation as well;
$$\left (\frac{ x}{ A^{(m)}(x)} \right )^{(-1)} = xA^{(m+1)}(x).$$
Explicitly (now treating $u_k$ as dummy indeterminates), $$ \begin{split} A^{(m)}_n(u_1,...,u_n) & = \sum_\text{partitions of $n$} M^{(m)}(e_1,e_2,...,e_n) u_1^{e_1}u_2^{e_3}\ldots u_n^{e_n}\\ & = \sum_\text{partitions of $n$} (-1)^{Se} \frac{1}{mn+1} \frac{(mn+Se)!}{(mn)!e_1!e_2!\ldots e_n!} u_1^{e_1}u_2^{e_3}\ldots u_n^{e_n}. \end{split} $$ A combinatorial reciprocity applies with $m$ being any integer and $$ \begin{split} M^{(m)}(e_1,e_2,...,e_n) & = (-1)^{Se} \frac{1}{mn+1} \frac{(mn+Se)!}{(mn)!e_1!e_2!\ldots e_n!} \\ & = \frac{1}{mn+1} \frac{(-mn-1!}{(-mn-1-Se)!e_1!e_2!\ldots e_n!}. \end{split} $$ The particular case with all the indeterminates for a fixed $m$ nulled, i.e., set to $0$, except for $u_1$ gives the coefficient of $A_n^{(m)}(u_1,0,0,...)$, the diagonal coefficient of $A_n^{(m)}(u_1,...,u_n)$, as the signed generalized Fuss-Catalan number sequences
$$FC^{(m)}_n = (-1)^n \frac{1}{mn+1} \binom{(m+1)n}{n} = \frac{1}{mn+1} \binom{-mn-1}{n},$$
so then the $A^{(m)}(x)$ become $FC^{(m)}(x)$, the o.g.f.s for the signed (m)-Fuss-Catalan number / diagonal (m)-associahedra sequence.
Again, $m$ can be any integer, with $FC_n^{(-1)}(x) = \sum_{n \ge 0}FC_n^{(-1)}x^n = 1 - x$ and $m=1$ and $-2$ giving variants of the Catalan numbers (A000108).
Dual results apply to the noncrossing partitions / parking function partition polynomials of OEIS A134264 and their diagonals, the Fuss-Narayana sequences; the Lagrange inversion partition polynomials for e.g.f.s of A134685; and the refined Eulerian partition polynomials of A145271, all related to compositional and multiplicative inversion.
Question: What other families of partition polynomials display similar sampling properties?
Since the associahedra polynomial $xA(x)$ gives compositional inverses, the lowering operation seems something of a slight of hand veiling the appearance of the full polynomial, but the compositional inversion can be achieved several ways--with the classic Lagrange inversion formula, a fixed point equation, an iterated Lie derivative, a Laplace transform method, Sheffer binomial polynomial formalism and matrix inversion, and even geometrically with reflection through the line $y=x$ or the envelope method and the Legendre-Fenchel transform.,
