The inverse $Z$-polynomial of a matroid
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915368573861888 |
|---|---|
| author | Gao, Alice L. L. Ruan, Xuan Xie, Matthew H. Y. |
| author_facet | Gao, Alice L. L. Ruan, Xuan Xie, Matthew H. Y. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The inverse $Z$-polynomial of a matroid Gao, Alice L. L. Ruan, Xuan Xie, Matthew H. Y. Combinatorics 05B35, 52B40, 05A20 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. |
| title | The inverse $Z$-polynomial of a matroid |
| topic | Combinatorics 05B35, 52B40, 05A20 |
| url | https://arxiv.org/abs/2507.01332 |