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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.05825 |
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Table of 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.