If I evaluate substitution of an infinite set of indeterminates $(c_1,c_2,c_3,\cdots)$ into the infinite set of refined Eulerian polynomials $[E]$ of OEIS A145271, I obtain the Taylor series coefficients at the origin of the compositional inverse
$f^{(-1)}(x) = x + E_1(c_1) \frac{x^2}{2!} + E_2(c_1,c_2) \frac{x^3}{3!} + \cdots = x + b_1 \frac{x^2}{2!} + b_2 \frac{x^3}{3!} + \cdots$
of the formal Taylor series, or exponential generating function
$f(x) = x + a_1 \frac{x^2}{2!} + a_2 \frac{x^3}{3!} + \cdots $
that satisfies
$g(x) = 1 + c_1x + c_2 \frac{x^2}{2!} + c_3 \frac{x^3}{3!} + \cdots = \frac{1}{f'(x)} = \frac{1}{1 +a_1x+ a_2 \frac{x^2}{2!} + a_3 \frac{x^3}{3!} + \cdots} .$
Denote this transformation by
$[E][c] = [b]$.
The same results issue from the following substitution into the classic Lagrange inversion polynomials $[L]$ of A134685:
$[L][a] = [b].$
In addition, if we define the reciprocal polynomials $[P]$ of A133314 by
$\frac{x}{h(x)} = \frac{1}{1 + d_1x + d_2 \frac{x^2}{2!} + d_3 \frac{x^3}{3!} + \cdots} = 1 + P_1(d_1) + P_2(d_1,d_2) \frac{x^2}{2!} + P_3(d_1,d_2,d_3) \frac{x^3}{3!} + \cdots$
(which turn out to be the refined Euler characteristic polynomials of the permutahedra), then
$[P][c] = [a]$.
We also have
$[P]^2 = [I] = [L]^2$,
where $[I]$ is the identity transformation under substitution.
Consequently,
$[L][a] = [L][P][c] = [b] = [E][c],$ so
$[c] = [P][L][b] = [P][L] [E] [c],$
implying the inverse under substitution of $[E]$ is
$[E]^{-1} = [P][L]$,
with the first few partition polynomials (with the $a_k$ now arbitrary, i.e., unrelated to those above)
$E_0^{(-1)} = 1$
$E_1^{(-1)} = a_1$
$E_2^{(-1)} = -a_1^2 +a_2$
$E_3^{(-1)} = 3 a_1^3 - 4 a_1a_2 + a_3$
$E_4^{(-1)} = -15 a_1^4 + 25 a_1^2 a_2 - 7 a_1a_3 - 4 a_2^2 + a_4$
$E_5^{(-1)} = 105 a_1^5 - 210 a_2 a_1^3 + 60 a_3 a_1^2 + 70 a_2^2 a_1 - 11 a_4 a_1 - 15 a_2 a_3 + a_5$
$E_6^{(-1)} = -945 a_1^6 + 2205 a_2 a_1^4 - 630 a_3 a_1^3 - 1120 a_2^2 a_1^2 + 126 a_4 a_1^2 + 350 a_2 a_3 a_1 - 16 a_5 a_1 + 70 a_2^3 - 15 a_3^2 - 26 a_2 a_4 + a_6.$
The sequence of sums of the coefficients of the monomials of the polynomials with all $a_k =-1$, in general, is
$(1,-A006351) = 1,-1,-2,-8,-52,-472,-5504, \cdots$,
with A006351 having one interpretation from the OEIS as the "Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon". The sequence (1,-A006351) is the set of Taylor series coefficients of the reciprocal of essentially the e.g.f. of A000311, the solution to "Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n".
The reduced form of $[E]^{-1}$ obtained with all the indeterminates set to $-t$ and the ultimate overall minus signs removed is A112493, which has combinatorial interpretations in the entry's reference "Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials" by Price and Sokal, and a variant of A112493 is A124324 (examine its diagonals), which has its own combinatorial models (run-sorting) presented in the three recent papers referenced in the entry.
Question: What are some combinatorial models for the refined inverse Eulerian polynomials $[E]^{-1}$?
A general note on associated combinatorics and analysis: $[L]$ of A134685 has a particularly simple model as $2n$ balls placed in bins and its reduced form A134991 is associated with phylogenetic trees, the tropical Grassmannians $G^{T}(2,n)$, the Whitehouse simplicial complexes, and stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. The partition polynomials of $[P]$, as already noted, label and enumerate geometrically distinct faces of the permutahedra and also can be modeled as weighted surjections. The o.g.f. analogue of this formulation is associated with the refined Narayana polynomials (noncrossing partitions) and their inverse partition polynomials, free probability theory, the Lagrange inversion associahedra polynomials, and characterizations of the scattering amplitudes of certain quantum fields, some involving again the tropical Grassmannians and phylogenetic trees. The reduced Eulerian polynomials A008292 are associated with descents and signs of permutations, properties of permutahedra, symmetric functions, and algebraic geometry/topology and characteristic classes. The sets of polynomials in this formulation as well as being rife with combinatorial models are associated with iterated Lie derivatives / vector fields, quadratic operads via Koszul duality characterized by multiplicative and compositional inversions, and algebraic geometry via compositional inversion.
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