The omega invariant of a matroid
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912980950581248 |
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| author | Fink, Alex Shaw, Kris Speyer, David E |
| author_facet | Fink, Alex Shaw, Kris Speyer, David E |
| contents | The third author introduced the $g$-polynomial $g_M(t)$ of a matroid, a covaluative matroid statistic which is unchanged under series and parallel extension. The $g$-polynomial of a rank $r$ matroid $M$ has the form $g_1 t + g_2 t^2 + \cdots + g_r t^r$. The coefficient $g_1$ is Crapo's classical $β$-invariant. In this paper, we study the coefficient $g_r$, which we term the $ω$-invariant of $M$. We show that, if $M/F$ is connected for every proper flat $F$ of $M$, and $ω(N)$ is nonnegative for every minor $N$ of $M$, then all the coefficients of $g_M(t)$ are nonnegative. We give several simplified versions of Ferroni's formula for $ω(M)$, and compute $ω(M)$ when $r$ or $|E(M)|-2r$ is small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The omega invariant of a matroid Fink, Alex Shaw, Kris Speyer, David E Combinatorics Algebraic Geometry 05B35 (Primary) 14T15, 52B45 (Secondary) The third author introduced the $g$-polynomial $g_M(t)$ of a matroid, a covaluative matroid statistic which is unchanged under series and parallel extension. The $g$-polynomial of a rank $r$ matroid $M$ has the form $g_1 t + g_2 t^2 + \cdots + g_r t^r$. The coefficient $g_1$ is Crapo's classical $β$-invariant. In this paper, we study the coefficient $g_r$, which we term the $ω$-invariant of $M$. We show that, if $M/F$ is connected for every proper flat $F$ of $M$, and $ω(N)$ is nonnegative for every minor $N$ of $M$, then all the coefficients of $g_M(t)$ are nonnegative. We give several simplified versions of Ferroni's formula for $ω(M)$, and compute $ω(M)$ when $r$ or $|E(M)|-2r$ is small. |
| title | The omega invariant of a matroid |
| topic | Combinatorics Algebraic Geometry 05B35 (Primary) 14T15, 52B45 (Secondary) |
| url | https://arxiv.org/abs/2411.19521 |