This question is motivated by

1. https://mathoverflow.net/questions/255757/why-do-combinatorial-abstractions-of-geometric-objects-behave-so-well
2. [The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
](https://arxiv.org/abs/1712.01250)

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

+ 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

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

- [The Hodge theory of Soergel bimodules](https://arxiv.org/abs/1212.0791), hinting its relation with higher category theory.

- [The Kazhdan-Lusztig polynomial of a matroid
](https://arxiv.org/abs/1412.7408), defining matroid analogue of KL polynomials.

- [The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
](https://arxiv.org/abs/1712.01250). As Sam pointed out in the comment, this paper does a great job collecting many different examples.