A principle for converting Lindstr\"om-type lemmas to Stembridge-type theorems, with applications to walks, groves, and alternating flows

TL;DR

This paper establishes a general principle connecting Lindström-type lemmas to Stembridge-type theorems, extending their applicability to walks, groves, and alternating flows, with implications for combinatorial and algebraic structures.

Contribution

It introduces a unifying framework that translates Lindström-type determinant relations into Stembridge-type Pfaffian relations, broadening the scope of combinatorial identities.

Findings

Generalization of Lindström's lemma to walks on directed graphs

Conversion of determinant relations to Pfaffian relations

Applications to groves and alternating flows in combinatorics

Abstract

We prove that Fomin's generalization of Lindstr\"om'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 Lindstr\"om-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.