A direct proof of well-definedness for the polymatroid Tutte polynomial
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910361381240832 |
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| author | Guan, Xiaxia Jin, Xian'an |
| author_facet | Guan, Xiaxia Jin, Xian'an |
| contents | For a polymatroid $P$ over $[n]$, Bernardi, Kálmán and Postnikov [\emph{Adv. Math.} 402 (2022) 108355] introduced the polymatroid Tutte polynomial $\mathscr{T}_{P}$ relying on the order $1<2<\cdots<n$ of $[n]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that $\mathscr{T}_{P}$ is equivalent to another polynomial that only depends on $P$. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A direct proof of well-definedness for the polymatroid Tutte polynomial Guan, Xiaxia Jin, Xian'an Combinatorics For a polymatroid $P$ over $[n]$, Bernardi, Kálmán and Postnikov [\emph{Adv. Math.} 402 (2022) 108355] introduced the polymatroid Tutte polynomial $\mathscr{T}_{P}$ relying on the order $1<2<\cdots<n$ of $[n]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that $\mathscr{T}_{P}$ is equivalent to another polynomial that only depends on $P$. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial. |
| title | A direct proof of well-definedness for the polymatroid Tutte polynomial |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.05825 |