Inverse Kazhdan--Lusztig polynomials of fan matroids

TL;DR

This paper derives explicit formulas and generating functions for inverse Kazhdan--Lusztig and Z-polynomials of fan matroids, revealing their combinatorial properties and log-concavity of coefficients.

Contribution

It provides explicit formulas and generating functions for these polynomials specifically for fan matroids, expanding understanding of their combinatorial structure.

Findings

Explicit formulas for inverse Kazhdan--Lusztig polynomials of fan matroids.

Generating functions for inverse Z-polynomials of fan matroids.

Coefficients of inverse Kazhdan--Lusztig polynomials are log-concave with no internal zeros.

Abstract

The inverse Kazhdan--Lusztig polynomial of a matroid was introduced by Gao and Xie, and the inverse $Z$-polynomial of a matroid was introduced by Ferroni, Matherne, Stevens, and Vecchi. In this paper, we study these two polynomials for fan matroids, a family of graphic matroids associated with fan graphs. We first derive the generating functions for the inverse Kazhdan--Lusztig polynomials of fan matroids using their recursive definition, and then deduce the explicit formulas of these polynomials therefrom. For the inverse $Z$-polynomials of fan matroids, we obtain their generating functions using a parallel generating function approach, and further derive their explicit expansions based on these generating functions. Additionally, we provide alternative proofs for the above generating functions using the deletion formulas for inverse Kazhdan--Lusztig and inverse $Z$-polynomials. As an application of the explicit formula for inverse Kazhdan--Lusztig polynomials, we prove that the coefficients of the inverse Kazhdan--Lusztig polynomial of the fan matroid form a log-concave sequence with no internal zeros.