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Convolution formulas for multivariate arithmetic Tutte polynomials

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Tianlong Ma, Xian'an Jin, Weiling Yang

Combinatorics · Submitted 7 Oct 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids.

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arXiv ID2310.04659 (version 1)
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CitationTianlong Ma, Weiling Yang, Xian'an Jin. "Convolution formulas for multivariate arithmetic Tutte polynomials." arXiv:2310.04659 (2023). https://pith.science/paper/MPYC4SDS
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The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.

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s0 The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. background 1.00 0.07 0.05 asserted 1.00
s1 In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. contribution 0.99 0.96 0.64 asserted 1.00
s2 In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. result 0.93 0.93 0.32 asserted 1.00
s3 Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. result 0.64 0.85 0.20 asserted 1.00
s4 The proofs presented here are significantly shorter than the previous ones. interpretation 0.63 0.75 0.07 asserted 1.00
s5 In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid. result 0.80 0.96 0.14 asserted 1.00

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Original abstract

The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.

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