The inverse $Z$-polynomial of a matroid
Abstract
Motivated by the -polynomials of matroids, Ferroni, Matherne, Stevens, and Vecchi introduced the inverse -polynomial of a matroid. In this paper, we prove several fundamental properties of the inverse -polynomial, including non-negativity and multiplicativity, and show that it is a valuative invariant. We also provide explicit formulas for the inverse -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 -polynomial, we conjecture that the coefficients of the inverse -polynomial are unimodal and log-concave.
Cite
@article{arxiv.2507.01332,
title = {The inverse $Z$-polynomial of a matroid},
author = {Alice L. L. Gao and Xuan Ruan and Matthew H. Y. Xie},
journal= {arXiv preprint arXiv:2507.01332},
year = {2025}
}