Looking for references for one or more of the following four sets of partition polynomials 1a) through 4a), particularly those which present geometric / topological combinatorial interpretations.
Taking the logarithm of an exponential generating function of moments (or applying OEIS https://oeis.org/[A127671][1] / A263634 ) gives the formal cumulants of classical probability theory whereas taking the shifted reciprocal of the compositional inverse of the ordinary generating function of the moments (or applying A350499, $[N^{(-1)}]$ below, see this MO-Q) gives the formal cumulants of free probability theory. They are interlinked by the analysis below.
For $n = 1,2,3,...$, the infinite set of mutually commuting indeterminates $u_n$, the monomials $M_{n}(e_1,...,e_n) = u_1^{e_1}u_2^{e_2}\cdots u_n^{e_n}$ for the set $\lambda$ of all partitions $(e_1,e_2,...,e_n)$ of $n$ with $n = \sum_{k=1}^n k e_k$, the sum of exponents $Se = \sum_{k=1}^n e_k$, and the multinomial coefficient $\binom{a}{b_1,b_2,...,b_n} = \frac{a!}{b_1!b_2!\cdots b_n!}$,
1a) $[LA]$, the log associahedra,
$$LA_n = \sum_{\lambda} \binom{-n}{Se} \binom{Se}{e_1,e_2,...,e_n} M_{n}(e_1,...,e_n)$$
$$= \sum_{\lambda} (-1)^{Se} \binom{n+Se-1}{Se} \binom{Se}{e_1,e_2,...,e_n} M_{n}(e_1,...,e_n)$$
$$= \sum_{\lambda} (-1)^{Se} \frac{(n+Se-1)!}{(n-1)!} \binom{1}{e_1,e_2,...,e_n} M_{n}(e_1,...,e_n)$$
2a) $[LN]$, the log noncrossing partitions / parking functions (refined Narayana polynomials)
$$LN_n = \sum_{\lambda} \binom{n}{Se} \binom{Se}{e_1,e_2,...,e_n} M_{n}(e_1,...,e_n)$$
$$ = \sum_{\lambda} \frac{n!}{(n-Se)!} \binom{1}{e_1,e_2,...,e_n} M_{n}(e_1,...,e_n)$$
3a) $[ILA] = [LA]^{-1}$, the inverse log associahedra,
$$ILA_n = \sum_{\lambda} (-1)^{Se} \frac{(n+1)^{Se} }{n+1} \binom{1}{e_1,e_2,...,e_n} \binom{1}{2^{e_2},3^{e_3},...,n^{e_n}} M_{n}(e_1,...,e_n)$$
4a) $[ILN] = [LN]^{-1}$. the inverse log noncrossing partitions
$$ILN_n = \sum_{\lambda} (-1)^{n} \frac{(-n+1)^{Se} }{-n+1} \binom{1}{e_1,e_2,...,e_n} \binom{1}{2^{e_2},3^{e_3},...,n^{e_n}} M_{n}(e_1,...,e_n).$$
(Commonly, in references on symmetric functions, something like $ \mu^{\lambda}$ or $\mu_{\lambda}$ denotes the factor $\binom{1}{e_1,e_2,...,e_n} $ and $1/z^{\lambda}$ or $z_{\lambda}^{-1}$ denotes $\mu_{\lambda} \binom{1}{2^{e_2},3^{e_3},...,n^{e_n}}$.)
Examples:
$LA_0 = LN_0= ILA_0 = ILN_0=1$ and
1b) $[LA]$:
$LA_1 = -u_1$
$LA_2 = 3 u_1^2 - 2u_2$
$LA_3 = -10 u_1^3 + 12 u_1 u_2 - 3 u_3$
$LA_4 = 35 u_1^4 - 60 u_1^2 u_2 + 10 u_2^2 + 20 u_1 u_3 - 4 u_4,$
2b) $[LN]$:
$LN_1 = u_1$,
$LN_2 = u_1^2 + 2u_2$,
$LN_3 = u_1^3 + 6 u_2 u_1 + 3 u_3$,
$LN_4 = u_1^4 + 12 u_2 u_1^2 + 12 u_3 u_1 + 6 u_2^2 + 4 u_4.$
3b) $[ILA]$:
$ILA_1 = -u_1$
$ILA_2 = (3u_1^2 - u_2)/2!$
$ILA_3 = (- 16 u_1^3 + 12 u_1 u_2 -2u_3 ) / 3! $
$ILA_4 = (125 u_1^4 - 150 u_2 u_1^2 + 40 u_3 u_1 + 15 u_2^2 - 6 u_4) / 4!,$
4b) $[ILN]$:
$ILN_(u_1) = u_1$,
$ILN_2(u_1,u_2) = (2u_2-2u_1^2)/2!$,
$ILN_3 = (4 u_1^3 - 6 u_2 u_1 + 2u_3)/3!$,
$ILN_4 = (-27 u_1^4 + 54 u_2 u_1^2 - 24 u_3 u_1 - 9 u_2^2 + 6 u_4)/4!.$
Origins and properties:
By inverse set of partition polynomials I mean that substituting the polynomials of the inverse set (on the right) as the indeterminates of the direct set (on the left) gives the substitutional identity; e.g., symbolically, globally,
$$[LN][ILN] =[LN] [LN]^{-1} = [I],$$
and locally, for $n \geq 0$,
$LN_n(u_1,u_2,...,u_n) |_{u_k \to ILN_k(u_1,...,u_k)} = LN_n(ILN_1(u_1),...,ILN_n(u_1,...,u_n)) = u_n$.
Formulating these results in terms of ordinary generating functions (o.g.f.s) / power series reveals numerous identities (and suggests generalizations).
Compositional inversion / series reversion of o.g.f.s $O(x) = x \;( 1 + u_1 x^2 + u_2 x^2 + \cdots)$ can be achieved either via the associahedra partition polynomials $A_n(u_1,...,u_n)$ of A133437 / A111785 or the noncrossing partitions polynomials / parking functions polynomials of A134264 $N_n(b_1,...,b_n)$ with $b_n =R_n(u_1,...,u_n)$ with $[R]$ the set of reciprocal PartPs giving the shifted multiplicative inverse $x/O(x) = 1 + \sum_{n \geq 1} R_n(u_1,...,u_n) x^x$.
With the o.g.f.s $P(t) = 1 + \sum_{n \geq 1} P_n(u_1,...,u_n) t^n$ for $P = A,N,LA,LN,ILA$ or $ILN$
$$LA(t) = \frac{d}{dt} \ln(A(t))$$
and
$$LN(t) = \frac{d}{dt} \ln(N(t)),$$
so $[LA]$ and $[LN]$ can be expressed as the Faber polynomials of A263916, the classic polynomials of symmetric function theory, in particular, the Newton identities, through which the elementary symmetric polynomials / functions and complete homogeneous symmetric polynomials / functions can be expressed in terms of the power sum polynomials.
Equivalently, as sketched in A263634 (cf. also A127671), $LA_n(u_1,...,u_n)$ and $LN_n(u_1,...,u_n)$ are the coefficients of the differential part of the raising operator for the Appell polynomials
$$\bar{A}_n(x) = (\bar{A}_.(u_1,...,u.)+x)^n = \sum_{k=0}^n \binom{n}{k} \bar{A}_k(u_1,...,u_k) x^{n-k} = \sum_{k=0}^n \binom{n}{k} k!A_k(u_1,...,u_k) x^{n-k} $$
and
$$\bar{N}_n(x) = (\bar{N}_.(u_1,...,u.)+x)^n =\sum_{k=0}^n \binom{n}{k} \bar{N}_k(u_1,...,u_k) x^{n-k} = \sum_{k=0}^n \binom{n}{k} k!N_k(u_1,...,u_k) x^{n-k} $$
That is, with $D_x = d/dx$ and
$$R_A = x + LA(D_x)$$
and $$R_N = x + LN(D_x),$$
then
$$\bar{A}_n(x) = R_A \; \bar{A}_{n-1}(x) = R_A^n \; 1$$
and
$$\bar{N}_n(x) = R_N \; \bar{N}_{n-1}(x) = R_N^n \; 1$$
with
$$\bar{A}_n(0) = (\bar{A}.(u_1,...,u.)+0)^n = (\bar{A}.(u_1,...,u.))^n = \bar{A}_n(u_1,...,u_n) = n!A_n(u_1,...,u_n) $$
and
$$\bar{N}_n(0) = n!N_n(u_1,...,u_n).$$
In addition, I can show, several ways, that
$$A(x) = \frac{1}{N^{(-1)}(x)}$$
and
$$N(x) = \frac{1}{A^{(-1)}(x)} $$
where $[N^{(-1)}]=[N]^{-1}$ is the set of ParPs of A350499, for which
$$[N][N^{(-1)}] = [N][N]^{-1}= [I],$$
the identity under substitution, and
$$[A^{(-1)}]= [N]^{-1}[R],$$
is the set of ParPs of A355201.
Then the Appell Sheffer finite operator calculus gives the raising operators for $\bar{N}_n^{(-1)}(x)$ and $\bar{A}_n^{(-1)}(x) $ as
$$R_{N^{(-1)}} = x - LA(D_x)$$
and
$$R_{A^{(-1)}} = x - LN(D_x).$$
These reciprocities are equivalent to the convolution identities (with the Kronecker delta $\delta_n = 0^n$, $\delta_0=1$)
$$\sum_{k=0}^n A_n(u_1,...,u_k)N_{n-k}^{(-1)}(u_1,...,u_{n-k}) = \delta_n$$
and
$$\sum_{k=0}^n A_n^{(-1)}(u_1,...,u_k)N_{n-k}(u_1,...,u_{n-k}) = \delta_n.$$
For easy reference, the coefficients of the monomials of
1c) $[A]$ are
$$(-1)^{Se} \frac{1}{n+1}\frac{(n+Se)!}{n!} \frac{1}{e_1!e_2!\cdots e_n!}$$
$$ =(-1)^{Se} \frac{1}{n+1}\binom{n+Se}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
$$ = \frac{1}{n+1}\binom{-n-1}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
2c) $[N]$,
$$\frac{1}{n+1} \frac{(n+1)!}{(n+1-Se)!} \frac{1}{e_1!e_2!\cdots e_n!}$$
$$= \frac{1}{n+1} \binom{n+1}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
3c) $[A^{(-1)}]$,
$$ \frac{-1}{n-1} \frac{(n-1)!}{(n-1-Se)!} \frac{1}{e_1!e_2!\cdots e_n!}$$
$$ = \frac{-1}{n-1} \binom{n-1}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
$$ = \frac{1}{-n+1} \binom{n-1}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
4c) $[N^{(-1)}] = [N]^{-1}$,
$$ - \; \frac{(n+Se-2)!}{(n-1)!} \frac{1}{e_1!e_2!\cdots e_n!}$$
$$ = - \; \frac{1}{n-1} \frac{(n+Se-2)!}{(n-2)!} \frac{1}{e_1!e_2!\cdots e_n!}$$
$$ = - \; \frac{1}{n-1} \binom{n+Se-2}{Se} \binom{Se}{e_1,e_2,...,e_n}$$
$$ = \frac{1}{-n+1} \binom{-n+1}{Se} \binom{Se}{e_1,e_2,...,e_n}.$$
Note the sign change $n$ to $-n$ transforms $A_n(u_1,...,u_n)$ to $A^{(-1)}_n(u_1,...,u_n)$ and $N_n(u_1,...,u_n)$ to $N^{(-1)}_n(u_1,...,u_n)$. There are different methods of deriving the polynomials which explain this simple change-in-sign duality (combinatorial reciprocity).
Some references in which I've already found subsets of $[LA],[LN],[ILA],[ILN]$ are
A) "Supplementary Exercises for Chapter 7 (symmetric functions) of Enumerative Combinatorics, vol. 2" by Richard P. Stanley (version of 4 June 2023), pg. 53, Exercise 133 d) contains $[LA]$ as well as does eqn. 2 on pg. 4 of "Parking functions and noncrossing partitions" by Stanley (1997).
B) "Jack polynomials" by Michel Lassalle (2009) , pg. 11, eqns. 4.7 and 4.8 contain $[ILA]$ and $[ILN]$.
C) "Symmetric functions" by Alain Lascoux (2001), eqn. 4.4.4 (f) on pg. 55 contains $[ILA]$ (his other manuscripts "Polynomials" and "Alphabet splitting" as well).
D) "Rational parking functions and Catalan numbers" by Drew Armstrong, Nicholas Loehr, and Gregory Warrington (2014), pg. 6, contains essentially $[ILA]$.
Note: These results can be extended to subsets of $[A],[A^{(1)}],[N],[N^{(-1}]$ indexed by $[A^{(m)}],[N^{(m)}]$ where $m$ is any integer. The factors $(pm 1 + \pm n)^p$ that occur in $[ILN]$ and $[ILA]$ could then likely be generalized to contain the characteristic polynomial of the m-extended Shi arrangement as discussed by Armstrong on pp. 22-3 of "Hyperplane Arrangements and Diagonal Harmonics", valuated at an integer.
