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.

中文翻译：

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将Lindström型引理转换为Stembridge型定理的原理，及其在步行，格罗夫和交替流中的应用

我们证明了Fomin对无环有向图上的路径到一般有向图上行走的路径的Lindström引理的推广也以相同的方式推广了Stembridge定理。此外，我们表明，只要一个操作族满足Lindström型行列式关系，一个相关的操作族就满足Stembridge型Pfaffian关系。我们将示例应用程序应用于Kenyon和Wilson的关于小树林的工作以及Talaska的关于交替流的工作。