# Combinatorics and geometry underlying a refined Pascal matrix/Newton identities

The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for transforming between complete homogeneous symmetric polynomials/functions and elementary symmetric polynomials/functions. Certain Koszul duals are related to this.

After reading Stanley's answer, I guess I need to rephrase the question to lean away from algebraic combinatorics towards geometric combinatorics. For example, the square of the lower triangular Pascal matrix is a convolution of binomial coefficients and also the face vectors of the hypersquares, or hypercubes. Another example, the refined associahedra partition face polynomials are combinations of averaged factorials that give a prescription for compositional inversion of o.g.f.s--it's the connection to the combinatorics of the faces--the geometry--that I would like emphasized.

On my blog "Shadows of Simplicity" I posted "Squaring Triangles" to explain to motivated high school students the relation between the algebraic and geometric combinatorics. Ideally, that is what I'm looking for here.

The algebraic combinatorics of the complementary reciprocal of a Taylor series/e.g.f. is governed by the antipode/refined Euler characteristic classes of the permutahedra or, equivalently, by surjective mappings, so I have an indirect geometric combinatorial interpretation of 'scaled' versions of the Newton identities, but I'm looking for more direct interpretations.

What geometric structures are enumerated by the integer coefficients of these partition polynomials for conversion of an o.g.f. into a reciprocal o.g.f.?

One answer from A133314 is colored surjections, where the arrows mapping into an element of the image induce a linear order by color that are then permuted. The resulting partition polynomials of A263633 are then the coefficients of the e.g.f. enumerating these mappings, i.e. the scaled, signed face partition polynomials of the permutahedra divided by n!.

Since $$\frac{1}{1+a_1x+a_2x^2+\cdots} = \sum_{k\geq 0}(-1)^k(a_1x+a_2x^2+\cdots)^k,$$ the coefficient of $$a_1^{c_1}a_2^{c_2}\cdots x^n$$, where $$n=\sum ic_i$$, is just $$(-1)^k$$ times the multinomial coefficient $${k\choose c_1,c_2,\dots}$$, where $$k=\sum c_i$$.
• Does jog my memory though. My old notes are long gone, but the individual terms $g^n(x)$ have something to do with colored somethings? You would know better than I. I'll check Flajolet and Joyal, but I'm interested in all geometric interpretations. – Tom Copeland Sep 8 '20 at 2:47