Combinatorial meaning of Kazhdan-Lusztig-Stanley polynomial This question is motivated by

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*Why do combinatorial abstractions of geometric objects behave so well?

*The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials

Kazhdan-Lusztig-Stanley polynomials (KLS) are vast generalizations of the classical Kazhdan-Lusztig polynomials, whose special values have deep meaning in representation theory [1]. They also include the matroid analogue studied in the past ten years. (Aside: they also include general zeta functions.)
KLS bridge combinatorics and algebraic geometry. While the nonnegative coefficients of the KLS can be interpreted as the dimension of suitable cohomologies of certain perverse sheaves [2], it seems to be less transparent in pure combinatorial settings.
I hope to understand KLS more from its combinatorial perspective, without any interference by the geometrical side. However, the definition of KLS [2] is done algebraically on the deformed dual of the underlying poset, making its meaning less transparent.
Question

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*Why did combinatorialists consider KLS in their point of view?

*Any baby examples of posets whose KLS shows rich combinatorial information right away?

Remark: I have no background in combinatorics. Being aware of that KLS also generalizes (in some sense) many combinatorial invariants (h-vector, g-polynomials).. I'd hope the answer can be pedagogical, and show the easiest nontrivial example.
Related

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*Twisted Incidence Algebras and Kazhdan-Lusztig-Stanley Functions-[Brenti], in which a nonassociative algebra is naturally given.


*The Hodge theory of Soergel bimodules, hinting its relation with higher category theory.


*The Kazhdan-Lusztig polynomial of a matroid
, defining matroid analogue of KL polynomials.


*The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
. As Sam pointed out in the comment, this paper does a great job collecting many different examples.
 A: This is maybe more addressed at the comments and the discussion with Timothy Chow, but I just wanted to point out that, at least in a certain context, there is a very very "concrete" description of the $h$-vector. Say $\mathcal{P}$ is a simple (convex, full-dimensional) polytope in $\mathbb{R}^n$. Then let $\phi$ be a generic enough linear functional on $\mathbb{R}^n$. Use $\phi$ to orient the $1$-skeleton of $\mathcal{P}$: orient an edge $uv$ from $u$ to $v$ if $\phi(u) < \phi(v)$ (since $\phi$ is generic there will not be ties). Then if $h=(h_0,h_1,\ldots,h_n)$ is the $h$-vector of $\mathcal{P}$ (defined in the usual way as a transform of the $f$-vector), we have that
$$ h_i = \# (\textrm{vertices $v$ with indegree $=i$})$$
according to our orientation of the $1$-skeleton. So for instance this explains that the $h_i$ are positive, that $h_0+h_1+\cdots+h_n$ is the number of vertices; also we will have a $h_i=h_{n-i}$ symmetry which swaps indegree according to $\phi$ for outdegree according to $-\phi$, etc.
Incidentally, I don't know who to attribute this simple but nice perspective on the $h$-vector to; to me it is folklore.
EDIT: As Richard notes in the comments this perspective is the same as the idea of a line shelling for a simplicial polytope, which I guess was assumed by Schläfli in his proof of the Euler-Poincaré formula and formally established by Bruggesser and Mani.
