Abstract
We prove that Fomin’s generalization of Lindström’s lemma for paths on acyclic directed graphs to walks on general directed graphs also generalizes a theorem of Stembridge in the same way. Moreover, we show that whenever a family of operations satisfies a Lindström-type determinant relation, a related family of operations satisfies a Stembridge-type Pfaffian relation. We give example applications to Kenyon and Wilson’s work on groves and to Talaska’s work on alternating flows.
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Acknowledgements
The author would like to thank Pavlo Pylyavskyy for many helpful conversations throughout the research and writing process, including for alerting the author to the question of whether Stembridge’s theorem could be generalized along the lines of Fomin’s theorem, and for pointing out the work of Talaska on alternating flows.
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Biesel, O. A Principle for Converting Lindström-Type Lemmas to Stembridge-Type Theorems, with Applications to Walks, Groves, and Alternating Flows. Graphs and Combinatorics 37, 1229–1246 (2021). https://doi.org/10.1007/s00373-021-02310-z
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DOI: https://doi.org/10.1007/s00373-021-02310-z