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arXiv · 2507.01332

The inverse $Z$-polynomial of a matroid

Alice L. L. Gao·Xuan Ruan·Matthew H. Y. Xie
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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.

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Alice L. L. Gao, Xuan Ruan, Matthew H. Y. Xie. 2025-07-02. The inverse $Z$-polynomial of a matroid. https://arxiv.org/abs/2507.01332

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