The inverse $Z$-polynomial of a matroid

TL;DR

This paper studies the inverse Z-polynomial of matroids, proving key properties like non-negativity, multiplicativity, and valuativity, and provides explicit formulas for certain classes, also exploring their unimodality and log-concavity.

Contribution

It introduces the inverse Z-polynomial of matroids, establishes its fundamental properties, and derives explicit formulas for uniform and sparse paving matroids, including conjectures on coefficient behavior.

Findings

Proved non-negativity and multiplicativity of the inverse Z-polynomial.

Established the inverse Z-polynomial as a valuative invariant.

Derived explicit formulas for uniform and sparse paving matroids.

Abstract

Motivated by the $Z$-polynomials of matroids, Ferroni, Matherne, Stevens, and Vecchi introduced the inverse $Z$-polynomial of a matroid. In this paper, we prove several fundamental properties of the inverse $Z$-polynomial, including non-negativity and multiplicativity, and show that it is a valuative invariant. We also provide explicit formulas for the inverse $Z$-polynomials of uniform matroids and a broader class of matroids, namely sparse paving matroids, which include uniform matroids as a special case. Furthermore, we establish the unimodality and log-concavity of these polynomials in the case of sparse paving matroids. Based on the properties of the $Z$-polynomial, we conjecture that the coefficients of the inverse $Z$-polynomial are unimodal and log-concave.