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The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the simplicial dual to the permutohedra.

The refined Eulerian numbers (RENs) of A145271 are related analytically to the compositional inversion of functions and formal generating series and to flow fields generated by tangent vectors. The $n$-th row of RENs are the numerical coefficients of the expansion of $(g(x)\frac{d}{dx})^ng(x)$ in terms of the monomials in the derivatives of $g(x)$, i.e.,

$$g_k=\frac{d^k}{dx^k}g(x).$$

For example,

$$(g(x)\frac{d}{dx})^3g(x) = 1 g_0^1 g_1^3 + 4 g_0^2 g_1^1 g_2^1 + 1 g_0^3 g_3^1.$$

With $(\omega,x) = (f(x),f^{(-1)}(\omega))$ and $g(x) = 1/f^{'}(x)$,

$$\exp[t g(x)d/dx]x = \exp[td/d\omega]f^{(-1)}(\omega) = f^{(-1)}(t+\omega)=f^{(-1)}(t+f(x)).$$

Evaluated at the origin of $x$, this gives the compositional inverse

$$\exp[tg(x)d/dx] x |_{x=0}=f^{(-1)}(t).$$

Associations:

1) With $f(x)$ represented as a generic Taylor series about the origin, or formal e.g.f., the REN partition polynomials become those of A134685, the coefficients for the classic Lagrange inversion formula (LIF), which when reduced become the matrix of Ward numbers A134991 comprised of the f-vectors of the Whitehouse simplicial complex and enumerating phylogenetic trees. $g(x)$ is associated to face vectors of the permutohedra A019538.

2) With $f(x)$ represented as a power series or formal o.g.f., the partition polynomials of the associated LIF A133437 become the signed refined face polynomials of the Stasheff polytopes, or A-type associahedra and the reduced array becomes A086810 (cf. also A033282) enumerating dissections of polygons.

3) With $f(x)$ represented in terms of the coefficients of the power series of its shifted reciprocal $f(x)= x/h(x)$, the partition polynomials of the associated LIF A134264 give a refined enumeration of the noncrossing partitions of polygons, which reduces to the Narayana array A001263.

So, the refined Eulerian numbers have some connections to rather important classical combinatorial and analytic constructs.

The Adler ref in A145271 poses the question of combinatorial interpretations of the RENs. An interpretation in terms of the enumeration of rooted trees is depicted on p. 9 in the Mathemagical Forests ref in A145271.

Question: What are some other combinatorial interpretations of these refined Eulerian numbers?

Edit 3/4/2020:

Analytic relations between the Eulerian numbers (A008292) and the RENs (A145271) from relations to symmetric polynomials (and formal power series):

The ordinary generating function (o.g.f.) for the elementary symmetric polynomials in the indeterminates $x_k)$ is

$$E(x) = \prod_k (1 + x_k \cdot x) = 1 + e_1 x + e_2 x^2 + ....$$

The o.g.f. for the associated complete homogeneous symmetric polynomials is

$$H(x) = \frac{1}{\prod_k (1 - x_k \cdot x)} = 1 + h_1 x + h_2 x^2 + ....$$

The two are related by

$$ E(x) = 1/H(-x).$$

Define

$$B(x) = \int_0^x H(-t)dt = x - h_1 \frac{x^2}{2} + h_2 \frac{x^3}{3} + ... = \ln(1+u.x),$$

where umbrally $(u.)^n = u_n = h_{n-1}$ with $h_0 =1.$

Then

$$dB(x)/dx = H(-x) = 1/E(x),$$

and $$g(x) = (dB(x)/dx)^{-1} = E(x)$$

defines a vector field for generating the compositional inverse of $B$ by

$$ A(x) = B^{-1}(x) = \exp[x \cdot g(z)d/dz] \; z |_{z=0}$$

$$ = \exp[x \cdot E(z)d/dz] \; z |_{z=0}.$$

Now OEIS A133932 can be used to determine the formal e.g.f. for $A(x)$ in terms of the $h_n$, or A145271, in terms of the $e_n$ with $g_n = n! e_n$; and A263633 can be used to transform between the two sets $e_n$ and $h_n$ (see, e.g., Formal group laws and binomial Sheffer sequences). The coefficients of the monomials of A263633 for the recprocal of o.g.f.s can be determined explicitly from those of A133314 and A049019 for the reciprocal of e.g.f.s or by applying the Faa di Bruno formula directly for the reciprocal of an o.g.f.

The coefficients for the monomials of A133932 are explicitly given in that entry and therefore the coefficients within each row of A145271 can be determined using those for A263633.

The discussion above suggests how the Eulerian numbers, A008292, are a natural specialization of the RENs A145271:

The inverse of the bivariate e.g.f. $A_{Eul}(x)$ for A008292 is the same as for $B(x)$ above with $x_k=0$ for $k>2$, and the Riccati equation (an autonomous O.D.E., an evolution equation, which underlies soliton solutions for the KdV equation) satisfied by A008292

$$dA_{Eul}/dx = (1+x_1A_{Eul}(x))(1+x_2A_{Eul}(x))=E(A_{Eul}(x))$$

generalizes to

$$dA/dx = \prod_k (1+x_kA(x))= E(A(x))$$

for the e.g.f. of A145271.

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