A combinatorial problem arises in relating connected and disconnected Green functions associated to a "zero-dimensional" quantum field theory presented by Brezin, Itzykson, Parisi, and Zuber in "Planar Diagrams" (via Eqn. 31 on p. 42). The first few partition polynomials characterizing this problem are given in OEIS A338135 for the refined $2-$Narayana polynomials as
$G_2 = 1 g_2$
$G_4 = 1g_4 + 2 g_2^2$
$G_6 = 1g_6 + 6 g_2 g_4 + 5 g_2^3$
$G_8 = 1g_8 + 8 g_2 g_6 + 4 g_4^2 + 28 g_2^2 g_4 + 14 g_2^4$
$G_{10} = 1g_{10} + 10 g_2 g_8 + 10 g_4 g_6 + 45 g_2^2 g_6 + 45 g_2 g_4^2 + 120 g_2^3 g_4 + 42 g_2^5,$
and Brezin et al. assert $G_{2p}$ enumerates the number of ways $2p$ points on a circle can be connected by non-overlapping clusters of $2q$-plets, which from their multinomial formula can be identified as equivalent to the number of noncrossing partitions (NCP) enumerated and flagged by the nonvanishing partition polynomials $N_n[h_1,h_2,...,h_n]$ of A134264 or Dyck paths of A125181 with $h_0=1$ when $h_k$ for $k$ odd is replaced by zero. (The connection to A134264 implies that the row sums of the coefficients are the Fuss-Catalan numbers OEIS A001764 and that the 'main diagonal' contains the Catalan numbers, A000108. The reduced polynomials generated with all $h_k=t$, give the $2-$Narayana polynomials with the coefficients A108767, with other combinatorial interpretations.)
This Green function array, as well as A134264 for the NCPs, popped up just recently in "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" by Cachazo and Umbert, characterizing scattering amplitudes in certain quantum field theories. C & U give combinatorial interpretations in terms of NCP.
There is a cornucopia of combinatorial/algebraic/geometric constructs associated with all of these the OEIS arrays, but I want to focus on an analytic relation between the NCP and the Green function array that involves an iterative composition or iterated substitution with the NCP polynomials:
(COMP) $\;\;\;\;\;\hat{G}_n(c_1,c_2,...,c_n) = N_n[N_1(c_1),N_2(c_1,c_2),...,N_n(c_1,...,c_n)]= N_n^{(2)}(c_1,c_2,...,c_n).$
For example:
$N_1(u_1) = u_1,$
$N_2(u_1,u_2) = u_1^2 +u_2,$
$N_3(u_1,u_2,u_3) = u_3 + 3 u_2 u_1 + u_1^3.$
Then substituting $N_n$ for $u_n$ in the last equation and expanding gives
$N_3[N_1(c_1),N_2(c_1,c_2) ,N_3(c_1,c_2,c_3)] = (c_3 + 3 c_2 c_1 + c_1^3) + 3(c_1^2 +c_2)(c_1) + c_1^3$
$ = c_3 + 6c_1c_2 + 5c_1^3 = N_3^{(2)}(c_1,c_2,c_3),$
which is the third partition polynomial of the re-indexed Green function array
$\hat{G}_1 = c_1,$
$\hat{G}_2 = c_2 + 2 c_1^2,$
$\hat{G}_3 = c_3 + 6 c_1 c_2 + 5 c_1^3,$
$\hat{G}_4 = c_4 + 8 c_1 c_3 + 4 c_2^2 + 28 c_1^2 c_2 + 14 c_1^4.$
Edit, July 7 and 13, 2022: Start
Lying at the heart of the two algebraic methods of generating these polynomials--by the zeroing out or by the substitutions--are the following theorems that Peter Bala presented in 2008 in the formula section of the reduced array A108767 for the $2-$Narayana polynomials (with some change in notation);
Define a functional $I$ on formal power series of the form $h(x) = 1 +h_1x + h_2x^2 + ...$ by the following iterative process. Define inductively $F^{(1)}(x) = h(x)$ and $F^{(n+1)}(x) = h(x \cdot F^{(n)}(x))$ for $n \geq 1$. Then set $I(h(x)) = \lim_{n-\to \infty} F^{(n)}(x)$ in the x-adic topology on the ring of formal power series; the operator $I$ may also be defined by $I(h(x)) := \frac{1}{x}(x/h(x))^{(-1)}= f^{(-1)}(x)/x$, the compositional inverse, or series reversion, of $f(x)= x/h(x)$.
Forming the $m$-fold composition $I^{(m)}(h(x))$ and then replacing $x$ with $x^m$ produces the same result as $I(h(x^m))$.
The main theorem stems from a fixed point-iteration perspective on the deceivingly simple, basic compositional inversion identity
(INV) $\;\;\;\;\;x\; h(f^{(-1)}(x)) = f^{(-1)}(x)$,
and the connection to the NCPs is that
$f^{(-1)}(x) = x\; (1 + N_1(h_1)\; x + N_2(h_1,h_2)\; x^2 + \cdots).$
In fact, INV appears as eqn. 29 on p. 41 of Brezin et al.
The $N_n$ polynomials are central to free probability theory (FPT), and the analysis in Brezin et al. is closely related to the calculus of that theory as it relates to random matrix theory and the Cauchy-Hilbert and Stieltjes transforms. A variant of INV is a core equation of FPT that appears as eqn. 2.6 (the R-transform) on p. 10 of "Combinatorics of the categories of noncrossing partitions" by Chapoton and Nadeau in which the $2-$Narayana polynomials (A108767) are presented, NCPS are translated into trees, and composition of morphisms are presented as substitutions of trees. The R-transform in FPT defines the relation between free cumulants and moments, analogous to the exp-log definition for the formal cumulants and moments of classical probability theory. The NCP polynomials, or refined Narayana polynomials, $N_n(c_1,...,c_n)$, give the free moments in terms of the free cumulants $c_n$. (This last transformation may be called the R-transform as well, and there are variants of INV and the R-transform depending on whether the formulation is couched in terms of power series or Laurent series and on whether the series have a constant summand.)
These results generalize to further substitutions and correspondingly different sets of zeroed indeterminates--as Bala's n-fold composition formula implies--such as other than $h_{3k}$, or other than $h_{4k}$, and so on, generating the refined $m-$Narayana polynomials, reducing to the $m-$Narayana polynomials, reducing in turn to a Fuss-Catalan number sequence, explained by the o.g.f. of the aerated (i.e., with intervening zeros) m-Fuss-Catalan sequence being generated as the compositional inverse in x of $f(x) = x - x^{m+1}$ with $m=1$, the Catalan sequence.
End
Given all the associated combinatorial constructs, one wonders
Is there a direct combinatorial proof of the self-composition identity (COMP) for the noncrossing partition polynomials? (Not necessarily restricted to the use of NCPs---there are a multitude of interpretations of these polynomials, such as arcs, lattice paths, and trees as well as dissections of polygons, and some might be more advantageous than others.)
(These results can be couched also in terms of the associahedra partition polynomials of A133437 and the reciprocal polynomials $R_n$ generated as $1/h(x) = 1 + R_1(h_1) x + R_2(h_1,h_2) x^2 + \cdots$ and in terms of a more general, expansion coefficient array investigated by Schur, so there are lots of connections to be made. An equivalent e.g.f. formulation exists as well.)
