A recursive definition for the polymatroid Tutte polynomial
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915549359898624 |
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| author | Guan, Xiaxia Jin, Xian'an Yang, Weiling |
| author_facet | Guan, Xiaxia Jin, Xian'an Yang, Weiling |
| contents | The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for $\mathscr{T}_{P}$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for $\mathscr{T}_{P}$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A recursive definition for the polymatroid Tutte polynomial Guan, Xiaxia Jin, Xian'an Yang, Weiling Combinatorics The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for $\mathscr{T}_{P}$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for $\mathscr{T}_{P}$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently. |
| title | A recursive definition for the polymatroid Tutte polynomial |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.11046 |