The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of the linear term being unity. The first few are listed below in the Detail section.
Typically trees (various types) are conjured up to provide combinatorial illustrations / proofs of the inversion property of these polynomials. See, e.g., pages 9 and 10 of “Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials” by Price and Sokal and “Incidence Algebra Antipodes and Lagrange Inversion in One and Several Variables” by Haiman and Schmitt.
An exception to this is presented in Theorem 1 of “Moduli spaces in genus zero and inversion of power series” by McMullen, “which shows that the universal formula for inversion of power series is encoded in the stratification of moduli space” albeit the proof is via trees.
Another interpretation is simply placement of balls in bins as discussed below. See also, e.g., my “Short note on Lagrange inversion”, “Combinatorics of Higher Derivatives of Inverses” by Johnson, or “A176740: E.g.f. Lagrange inversion partition array” by Lang.
My question:
What other combinatorial constructs other than trees serve as models of the classic Lagrange inversion polynomials?
Details and additional remarks
Given the formal e.g.f / Taylor series
$$y = f(x) = a_1\; x + a_2 \; \frac{x^2}{2!} + a_3 \; \frac{x^3}{3!} + \cdots,$$
the formal compositional inverse about the origin is
$$x = f^{(-1)}(y) = p_1(a_1) \; y + p_2(a_1,a_2) \; \frac{y^2}{2!} + p_3(a_1,a_2,a_3) \; \frac{y^3}{3!} + \cdots \;.$$
The first few Lagrange inversion partition polynomials are
$p_1(a_1) = a_1^{-1}(1)$,
$p_2(a_1,a_2) = a_1^{-3} ( -a_2 ) $,
$p_3(a_1,a_2,a_3) = a_1^{-5} (3 \;a_2^2 - \;a_1a_3) $,
$p_4(a_1,a_2,a_3,a_4) = a_1^{-7} ( -15 \;a_2^3 + 10 \;a_1 a_2a_3 - a_1^2 a_4 )$,
$p_5(a_1,...,a_5) = a_1^{-9} ( 105 \; a_2^4 - 105 \; a_1 a_2^2 a_3 + 15 \; a_1^2 a_2 a_4 + 10 \; a_1^2 a_3^2 - a_1^3 a_5 ) $,
$p_6 (a_1,...,a_6)= a_1^{-11} ( -945 \; a_2^5 + 1260 \; a_1 a_2^3 a_3 - 280 \;a_1^2 a_2 a_3^2 - 210\; a_1^2 a_2^2 a_4 + 21 \; a_1^3 a_2a_5 + 35 \; a_1^3 a_3a_4 - a_1^4 a_6 ) .$
Letting $b_{m-1}= a_m$ and $b_0=a_1=1$ in each polynomial, the partitions / monomials / summands in the parentheses of $p_n$ become those of the partition of $n-1$ listed in Abramowitz and Stegun on page 831, and the number of partitions in $p_n$ is given by A000041(n-1). For example, for $p_6$ the monomial $280 \;a_1^2 a_2 a_3^2$ becomes $280 \;b_1b_2^2$ representing the partition $5 = 1+2+2$.
The coefficient of a monomial of $p_n$ of the form $ a_1^{e_1} a_2^{e_2} \cdots a_n^{e_n}$ is $(-1)^{n-1+e_1} \; \frac{(2(n-1)-e_1)!}{(2!)^{e_2}(e_2)!\;(3!)^{e_3}(e_3)! \cdots (n!)^{e_n}(e_n)!}$. The partition / monomial of $p_n$ with the factor $a_1^{e_1}$ satisfies $n-1 = e_1 +e_2 +e_3 +\cdots+ e_n$ and $2(n-1) = e_1 +2e_2 +3e_3 +\cdots+ ne_n$. The magnitude of the monomial $ a_1^{e_1} a_2^{e_2} \cdots a_n^{e_n}$ of $p_n$ corresponds to the number of ways $(2e_2 + 3e_3 + \cdots + n\; e_n) = 2(n-1)-e_1 $ labeled items can be separated without regard to order into $e_2$ bins of size $2$, $e_3$ bins of size $3$, ... , and $e_n$ bins of size $n$. E.g., the monomial $3 \; a_2^2$ of $p_3$ corresponds to the number of ways $4$ items may be grouped into two bins of equal size $2$, giving $\frac{4!}{2!^2\;2!} =3 $ ways of placement for which the linear order of the items is not important, corresponding to $(a,b)(c,d),(a,c)(b,d),$ and $(a,d)(b,c)$. In the same polynomial, $a_1a_3$ corresponds to the number of ways $3$ items may be distributed into one bin of size $3$, giving $\frac{3!}{3!1!} = 1$, corresponding to $(a,b,c)$.
Letting $a_1 =t$ and $a_n =1$ otherwise for the numerator polynomials generates the Ward polynomials of A134991 (mod signs), associated to several combinatorial models. (They are also a rotated version of A008299, the associated Stirling numbers of the second kind.)
Example 4.1 on page 395 of "The tropical Grassmannian" by Speyer and Sturmfels has a refinement of the numerical coefficients of $p_5$.
Edit 9/8/2023: Again, a reduced form of A134685 is A134991 for the Ward polynomials and these enumerate the faces of the tropical $G(2,n)$ Grassmannians (as noted in this MO-Q). Cachazo, Early, Guevara, and Mizera discuss on pgs. 13-14 of "Scattering Equations: From Projective Spaces to Tropical Grassmannians" connections between these tropical Grassmannians and the Feynman diagrams of $\phi^3$ quantum field theory. On the other hand, Lukowski, Parisi, and Williams assert on p. 35 of "The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron" that the f-vectors of $Trop^+$ $Gr_{2,n}$ are those of the associahedra, which are reduced vector versions of the more refined Lagrange inversion partition polynomials of A133437 for power series / o.g.f.s and so easily scale to the Lagrange inversion partition polynomials of A134685 for Taylor series / e.g.f.s. Can these models for the reduced polynomials be refined to include the full partition polynomials for Lagrange inversion?
