This question is motivated by

- 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

- 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, 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.

musthave some direct combinatorial meaning, then I think you can regard that as an open problem. $\endgroup$ – Timothy Chow Oct 2 at 12:57topologicallyand not combinatorially. But if you can find a convincingcombinatorialinterpretation of the Euler characteristic, then perhaps you can generalize that to a combinatorial interpretation of the h-vector. $\endgroup$ – Timothy Chow Oct 2 at 13:01