A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $\rho$-Removed Uniform Matroids

TL;DR

This paper introduces a new combinatorial formula for Kazhdan-Lusztig polynomials of -removed uniform matroids, utilizing skew Young Tableaux, which simplifies calculations and ensures positive integer coefficients.

Contribution

It provides the first combinatorial formula for these polynomials, extending known results for uniform matroids and improving computational manageability.

Findings

New combinatorial formula for -removed uniform matroids

Formula ensures positive integer coefficients

Simplifies computation compared to previous methods

Abstract

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of $\rho$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.