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On the Compatible Sets Expansion of the Tutte Polynomial

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Abstract

Kochol [6] gave a new expansion formula for the Tutte polynomial of a matroid using the notion of compatible sets, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas’s trivariate Tutte polynomials of matroid perspectives [10]. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol in [5] and [7] in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas’s activities expansion.

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Acknowledgements

The author thanks Christopher Eur for the problem suggestion and for providing helpful comments and references, and the anonymous reviewers for suggesting additional references and making the observation mentioned in Remark 2.

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  1. Harvard University, Cambridge, MA, 02138, USA

    Laura Pierson

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  1. Laura Pierson
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Correspondence to Laura Pierson.

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Communicated by Kolja Knauer.

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Pierson, L. On the Compatible Sets Expansion of the Tutte Polynomial. Ann. Comb. 28, 33–42 (2024). https://doi.org/10.1007/s00026-023-00657-z

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