Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.25254

Inverse Kazhdan--Lusztig polynomials of fan matroids

Alice L. L. Gao·Ya-Xing Li·Yun Li
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alice L. L. Gao, Ya-Xing Li, Yun Li. 2026-05-12. Inverse Kazhdan--Lusztig polynomials of fan matroids. https://arxiv.org/abs/2603.25254

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗
arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

math.CO↗