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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.11666 |
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| _version_ | 1866913574595592192 |
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| author | Byrne, Eimear Fulcher, Andrew |
| author_facet | Byrne, Eimear Fulcher, Andrew |
| contents | We describe a construction of the Tutte polynomial for both matroids and $q$-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct $q$-analogue of such partitions. We propose axioms of $q$-Tutte-Grothendiek invariance and show that this yields a $q$-analogue of Tutte-Grothendiek invariance. We establish the connection between the rank polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11666 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Invariants of Tutte Partitions and a $q$-Analogue Byrne, Eimear Fulcher, Andrew Combinatorics We describe a construction of the Tutte polynomial for both matroids and $q$-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct $q$-analogue of such partitions. We propose axioms of $q$-Tutte-Grothendiek invariance and show that this yields a $q$-analogue of Tutte-Grothendiek invariance. We establish the connection between the rank polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution. |
| title | Invariants of Tutte Partitions and a $q$-Analogue |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.11666 |